Zvonimir Janko (1932-2022)

outstanding Croatian mathematician


Professor Zvonimir Janko, Mathematical Institute, University of Heidelberg, Germany, at the University of Zagreb, Faculty of Electrical Engineering and Computing, November 26, 2007, at a scientific meeting organised on the occasion of his 75th birthday by his students and collaborators in Croatia.


Contents of this web page

Classification Theorem

Let us state the famous Classification Theorem for finite simple groups.


Theorem. Each finite nonabelian simple group is either of Lie type, or alternating group, or one of the following 26 sporadic groups:

  • M11, M12, M22, M23, M24
  • J1, J2, J3, J4
  • Hs
  • Co1, Co2, Co3
  • He
  • Mc
  • Suz
  • M(22), M(23), M(24)'
  • Ly
  • Ru
  • ON
  • F5, F3, F2, F1

Definition of the group, with a few examples and applications. More advanced, for beginning graduates: J.S. Milne, Group Theory, 2007, 121 pp.

Daniel Gorenstein, the first person to put forward a plan for classifying all the finite simple groups, started the introduction to his book "The Classification of Finite Simple Groups" with the following words:

Never before in the history of mathematics has there been an individual theorem whose proof has required 10,000 journal pages of closely reasoned argument. Who could read such a proof, let alone communicate to others? But the classification of all finite simple groups is such a theorem - its complete proof, developed over a 30-year period by about 100 group theorists, is the union of some 500 journal articles covering approximately 10,000 printed pages.

A few pages later Gorenstein continues:

For it is almost impossible for the uninitiated to find the way through the tangled proof without an experienced guide: even the 500 papers themselves require careful selection from among some 2,000 articles on simple groups theory, which together include often attractive byways, but which serve only to delay the journey.

It is therefore not surprising that the Classification theorem, the formulation of which is rather short, is also called enormous theorem due to its enormously long proof. Moreover, there is not a single expert capable to understand the proof of the Classification Theorem in its entirity.


Professor Zvonimir Janko delivered an invited lecture about his results concerning sporadic groups at the International Congress of Mathematicians in Nice, France, in 1970. Behind him on the right is professor Vladimir Devidé, University of Zagreb, his PhD advisor.


Janko groups

The sporadic groups have been discovered already in the 19th century by Émile Léonard Mathieu (1835-1890). Mathieu goups M11 and M12 have been discovered in 1861, and M22, M23, and M24 in 1873.

It took more than 90 years after the discovery of the last Mathieu group (and more than a century after the discovery of the first Mathieu group) until professor Zvonimir Janko, then a young mathematician at the age of 32, constructed a new sporadic group in 1964, now called J1 in his honour. It has 175,650 elements.

Professor Zvonimir Janko in 1964, at the age of 32, when he discovered J1

The discovery of J1 in 1964 (published in 1966) was a tremendous surprise in mathematical community. It had been "received as a sensation by the specialists in group theory", see [Held, p1]. This discovery has launched a modern theory of sporadic groups. Zvonimir Janko at that time worked in Australia, at The Australian National University (his name is included within Notable past staff) in Canberra, capital of Australia. Subsequently Janko discovered three more sporadic groups:

  • J2 (with Hall, also denoted HJ and called Hall-Janko group), in 1966,
  • J3 (with G. Higman and McKay, also called Higman-Janko-McKay group, HJM), also in 1966, and
  • J4 (with Norton and Parker) in 1975.

In 1967 Higman and Sims constructed sporadic group HS. John Conway constructed three sporadic groups, Co1 and Co2 in 1968, and Co3 in 1968.

Dieter Held constructed a sporadic group now called He (with G. Higman and Mc Kay), J.E. McLaughlin and Michio Suzuki constructed respective sporadic groups Mc and Su in 1968.

Fischer constructed groups M(22), M(23) and M(24)' in 1969. Richard Lyons and Charles Sims constructed Ly in 1970, Rudvalis, Conway and Wales constructed Ru in 1972. The same year Michael O'Nan and Charles Sims discovered ON.

In 1974 the following three sporadic groups have been discovered: F5 by John Thompson and Stephen Smith, F3 by Koichiro Harada, Simon Norton and Scmith, and F1 by Bernd Fischer and Robert Griess. The sporadic group F1 is the largest one, better known as the Monster group. It has approximately 8x1053 elements, or precisely


elements. The second largest one is F2, called the baby Monster, which has


elements. The fourth largest sporadic group is J4, with order equal to


It should be stressed that Zvonimir Janko not only discovered the first sporadic group in the 20th century (J1 in 1964), but also the last one (J4 in 1975). So Janko somehow opened and closed the quest for all sporadic groups in the 20th century. It resulted in the discovery of altogether 21 sporadic groups in the 20th century.

The fascinating quest for new sporadic groups has resulted, besides four Janko groups, in the discovery of 17 of them in the period from 1967 to 1975: one jointly by Donald Higman and Charles Sims; one each by Dieter Held, Richard Lyons, Jack McLaughlin, Michael O'Nan, Michio Suzuki, and Arunas Rudvalis; three by John Conway; and seven by Bernd Fischer, with important contributions by Robert Griess. The largest one, discovered by Fisher and Griess, is well known as The Monster. In 1981 it has been proved by Simon Norton, Cambridge University, that there are no other sporadic groups than these 26 peculiar groups.

In January 1981 M. Aschbacher declared during a solemn session of the American Academy of Sciences that all finite simple groups are known. This meant that the Classification Theorem has been proved, and Janko's contribution to its proof was enormous. However, it turned out that Aschbacher's announcement was premature, since many mistakes have been found in the studies of the so called quasi-thin groups, and this has been finally settled in 1992, see Janko's survey paper [Janko], p 179.

The name of "sporadic groups" has been introduced by William Burnside in his monograph Theory of Groups of Finite Order, 1911, in which he indicated an exceptional nature of Mathieu groups: "These apparently sporadic simple groups would probably repay a closer examination than they have yet received." In this way he forsaw the avalanche of investigations which ensued only after 1964, when Janko discovered his J1.

To illustrate the "earthquake" in group theory which raised Janko's discovery of J1 we cite the following testimony from [Ćepulić] (translated from Croatian by D.Ž.). In 1997, during the celebration of 65th birthday of professor Janko in Mainz, Bertram Huppert, the author and coauthor (with Norman Blackburn) of the most extensive, encyclopaedic treatise and manual on finite groups, Endliche gruppen I, II, III, said roughly the following:

"There were a very few things that surprised me in my life. I experienced the Second World War. It could have been predicted that there will be war. I believe it will surprise you, and some of you may be shocked by what I am going to say. There were only the following two events that really surprised me: the discovery of the first Janko group and the fall of the Berlin Wall."

Professor Vladimir Ćepulić from the University of Zagreb was a witness of the following interesting event in Göttingen, which nicely illustrates the importance of this discovery.

In his investigation professor Janko exploited modular characters of groups, a theory developed in 40s and 50s of the 20th century by Richard Brauer, one of the most famous mathematicians. Professor Richard Brauer, a German emigree to the USA, was a visiting professor in the academic year 1964/65 at the Mathematical Institute in Göttingen, where I also participated as a stipendist of the Humboldt Foundation, and attended his lectures. Professor Brauer arrived to his first lecture after Christmas visibly excited, carring a piece of paper in his hand and saying: "I have received this mail from Zvonimir Janko from Australia, in which he informs me that using my theory of modular characters he found a new sporadic finite simple group!" ([Ćepulić], translated form Croatian by D.Ž.)

The name of Zvonimir Janko has been included twice into the Encyclopaedia Britannica Year Book with extensive presentations of his work written by Irving Kaplansky, a famous algebraist. See [Devidé].

Hall-Janko graph, source en.wikipedia.org,
(there are 90 outer vertices and 10 inner vertices)


Group J2 has 604 800 elements. It is the only one among four Janko groups to be a subgroup of the Monster. Therefore, it makes part of the so called "happy familly", a name coined by Robert Griess, see here. The "happy familly" is defined as the set of all sporadic groups that are proper subgroups of the Monster. There are 20 such groups. So, there are five groups which do not belong to the "happy family", and three of them are the Janko groups: J1, J3, andJ4. As we see, 26 sporadic groups are in some sense "exceptional" groups, and furthermore, the three indicated Janko groups are in some sense "exceptional" among sporadic groups!

Unlike the Fischer groups, Conway groups, and Mathieu groups, the Janko groups do not form a series and have little in common, see here.

In the construction of the Janko group J1 for the first time a computer in Canberra has been used in algebraic studies. This was the first Australian scientific computer. The group J1 has found applications in the physics of elementary particles. With the discovery of J1 Janko showed that a claim due to R. Brauer and J. Thompson about nonexistence of new simple groups was wrong (see Janko's biography in [Croatian biographical lexicon], p 323. Janko describes in his lively article [Janko, pp 176-177] how thanks to encouragement of his wife Zora to be persistent, he managed to find mistakes in the works of Thompson and Brauer. And these mistakes of great mathematicians brought him to discovery of J1 in 1964. That is why he calls J1 also Zora's group (Zorina grupa in Croatian, see in [Hanjs]), in honour to his wife. At that time Janko was a research fellow in Canberra, and this discovery immediately brought him the position of full professor at the University of Monach in Melbourne in 1965 at the age of 33, which meant that he skipped the usual three academic steps.

The group J2 has several geometric realizations, and according to his own words, see [Janko, p 177], the most beautiful is the one found by Jacques Tits from Paris.

After eight years of thorough investigations Janko managed to discover the group J4. It was discovered in Heidelberg, in May 21st, 1975, using the famous Thompson formula, and without any help of a computer (becuase it is impossible), see Janko's survey article [Janko, p 179].

Paul Halmos (1916-2006) in his well known book I have a photographic memory (Providence, 1987) provides photos of many outstanding mathematicians. Two of them are from Croatia: Zvonimir Janko and William Feller.



Professor Janko is an aggreable and communicative person.

Simple groups ballad

Believe it or not, Janko's groups have entered a ballad! Here is A Simple Ballad dealing with simple groups, due to anonymous author, and published by the American Mathematical Monthly, Nov. 1973 (provided on the web by Hubert Grassmann).



(Sung to the tune of "Sweet Betsy from Pike", [Midi file])

What are the orders of all simple groups?
I speak of the honest ones, not of the loops.
It seems that old Burnside their orders has guessed:
except of the cyclic ones, even the rest.

Groups made up with permutes will produce more:
For An is simple, if n exceedes 4.
Then, there was Sir Matthew who came into view
exhibiting groups of an order quite new.

Still others have come on the study this thing.
Of Artin and Chevalley now shall sing.
With matrices finite they made quite a list.
The question is: Could there be others they've missed?

Suzuki and Ree then maintained it's the case
that these methods had not reached the end of the chase.
They wrote down some matrices, just four by four,
that made up a simple group. Why not make more?

And then came up the opus of Thompson and Feit
which shed on the problem remarkable light.
A group, when the order won't factor by two,
is cyclic or solvable. That's what's true.

Suzuki and Ree had caused eyebrows to raise,
but the theoreticians they just couldn't faze.
Their groups were not new: if you added a twist,
you could get them from old ones with a flick of the wrist.

Still, some hardy souls felt a thorn in their side.
For the five groups of Mathieu all reason defied:
not A_n, not twisted, and not Chevaley.
They called them sporadic and filed them away.

Are Mathieu groups creatures of heaven or hell?
Zvonimir Janko determined to tell.
He found out what nobody wanted to know:
the masters had missed 1 7 5 5 6 0.

The floodgates were opened! New groups were the rage!
(And twelve or more sprouded, to great the new age.)
By Janko and Conway and Fischer and Held,
McLaughtin, Suzuki, and Higman, and Sims.

No doubt you noted the last don't rhyme.
Well, that is, quite simply, a sign of the time.
There's chaos, not order, among simple groups;
and maybe we'd better go back to the loops.


Remark 1. Found scrawled on a library table in Eckhart Library at the U. of Chicago; author unknown or in hiding. (See W. E. Mientka, Professor Leo Moser - Reflections of a Visit, American Mathematical Monthly 79 (1972), 609-614.)

Remark 2. The ballad was written before the discovery of the last sporadic group in 1975, that is, of the Janko group J4. In other words, the ballad is far from being finished.

Orders of all 26 sporadic simple groups, from the David Madore web page:

(spor-m11) 7920
(spor-m12) 95040
(spor-j1) 175560
(spor-m22) 443520
(spor-j2) 604800
(spor-m23) 10200960
(spor-hs) 44352000
(spor-j3) 50232960
(spor-m24) 244823040
(spor-mc) 898128000
(spor-he) 4030387200
(spor-ru) 145926144000
(spor-sz) 448345497600
(spor-on) 460815505920
(spor-co3) 495766656000
(spor-co2) 42305421312000
(spor-f22) 64561751654400
(spor-f5) 273030912000000
(spor-ly) 51765179004000000
(spor-f3) 90745943887872000
(spor-f23) 4089470473293004800
(spor-co1) 4157776806543360000
(spor-j4) 86775571046077562880
(spor-f24) 1255205709190661721292800
(spor-f2) 4154781481226426191177580544000000
(spor-f1) 808017424794512875886459904961710757005754368000000000


How was J1 discovered?


According to Mathematical Geneaology Project, the following eighteen mathematicians are PhD students of professor Zvonimir Janko, and he has 83 descendants (as of 2022):

  • Nita Bryce, Monash University 1969
  • Dean Crnkovic, University of Zagreb 1998
  • Terence Gagen, Australian National University 1967
  • Erwin Hess, Ruprecht-Karls-Universität, Heidelberg 1978
  • Syed Muhammed Husnine, Ruprecht-Karls-Universität, Heidelberg 1976
  • Elizabeta Kovač-Striko, Ruprecht-Karls-Universität Heidelberg 1975
  • Thomas Kölmel, Ruprecht-Karls-Universität, Heidelberg 1991
  • J. Lundgren Jr., The Ohio State University 1971
  • David Parrot, Monash University
  • Mario Osvin Pavcevic, University of Zagreb, 1996
  • Rolando Pomareda, The Ohio State University 1972
  • Kristina Reiss, Ruprecht-Karls-Universität Heidelberg 1980
  • Bujar Shita
  • Fredrick Smith, The Ohio State University 1972
  • Tran Van Trung, Ruprecht-Karls-Universität Heidelberg 1976
  • Sia Wong, Monash University, 1969
  • Kenneth Yanosko, The Ohio State University 1970
  • Ali Reza Zokaui, Ruprecht-Karls-Universität Heidelberg 1978

In fact, the number of his scientific descendants, direct and indirect, is much larger. In Croatia only there are about 20 of them.



Professor Zvonimir Janko is a member of Advisory Board of Glasnik Matematički,
a scientific journal issued by the Croatian Mathematical Society.


Professor Zvonimir Janko is a corresponding memberof the Croatian Academy of Sciences and Arts since 1995. He is a member of the Heidelberg Academy of Sciences since 1972. In 1970 the French Academy of Sciences decorated him with a medal for the discovery of his sporadic groups. Since 1998 he is a honorary citizen of Bjelovar, his native town.



In May 24, 2006, the following lecture was delivered at the Lighthill Institute of Mathematical Sciences in London:

Professor Mark Ronan, University of Illinois Chicago

This talk will describe the quest to find a complete list of all finite simple groups. These groups, the basic building blocks for all finite groups, were instrumental in Galois' work on the solvability of algebraic equations; and he himself discovered some important ones. More were exhibited in Jordan's great treatise in 1870, and further families emerged from Lie's work, as a result of the classification work by Killing and Cartan in the late nineteenth century. After the Second World War new families of simple groups were discovered, and there was enormous interest in finding a complete list. A way forward was found using work of Richard Brauer, and the great theorem of Walter Feit and John Thompson. They showed that every finite simple group whose order is not a prime number must contain elements of order 2, leading to some important subgroups that offered a method for completing the list.

To cut a long story short, while Thompson was advancing these new methods, Zvonimir Janko, a Croatian mathematician working in Australia, surprised the world with a very strange exceptional group. This was the first exception since Émile Mathieu discovered five beautiful groups of permutations in the nineteenth century, and it really set the cat among the pigeons. Further new exceptions came thick and fast, and they were dubbed "sporadic groups". The largest is called the Monster. This talk will explain how the Monster was discovered, and how it came to reveal strange connections between number theory and mathematical physics, sometimes called the moonshine connections.

Marc Ronan: Symmetry and the Monster, One of the Greatest Quests of Mathematics, Oxford University Press, 2006 (mathematicians involved in this book, including professor Janko): from the book review [PDF] by Robert L. Griess published in the Notices of the American Mathematical Society, February 2002:

... Zvonimir Janko, like Fischer, had very strong personal ideas about where to look for new simple groups, and he worked them quite hard. Janko's most successful theme was the so-called "O 2 extraspecial" hypothesis for centralizers of involutions. The flavor of Janko's program was more "internal group theory" rather than "external geometry". Fischer and Janko each found several new sporadic groups but by mining rather different parts of the group theory terrain. ...



Professor Zvonimir Janko with his wife Zora.

A short biography of Zvonimir Janko

Professor Zvonimir Janko was born in Bjelovar, Croatia, in 1932. He studied mathematics at the University of Zagreb. After graduation he was sent to work at a high school in a small town of Lištica (also known as Široki Brijeg - Wide Hill) in Bosnia and Herzegovina, where he was teaching physics.

There he had already started to publish his first scientific works. It is at that time that he learned about groups, and found it interesting that only four axioms in the definition are the starting point of enormous theory. It is amusing that he was receiving letters from editorial boards indicating "University of Listica" in his address! And the highest educational institution in the town was the high school where he worked. Professor Janko's wife Zora is from Lištica.

He earned his PhD at the University of Zagreb in 1960 under (formal) supervision of professor Vladimir Devidé. The title of the thesis was Dekompozicija nekih klasa nedegeneriranih Rédeiovih grupa na Schreierova proširenja (Decomposition of some classes of nondegenerate Rédei Groups on Schreier extensions), in which he solved a problem proposed by an oustanding Hungarian mathematician Laszlo Rédei. Professor Stanko Bilinski (PDF) was the head of the PhD exam committee.

Until 1962 Janko alreday had about ten published journal articles, thus satisfying conditions for the position of assistant professor of mathematics at the University of Zagreb. Despite excellent results of a young talented mathematician, Janko was not able to find an adequate job in the communist Yugoslavia. His only "fault" was his particiaption in students' visit to the grave of the 19th century Croatian politician Dr. Ante Starčević, for which he was draconcially punished by two years of suspension of his studies. This is why his studies were prolonged for two years, from 1950 til 1960.

He applied thirteen times for the position of Research Fellow or Assistent Professor in ex-Yugoslavia, but in vain (even in Tuzla in Bosnia and Herzgovina.) Therefore Janko decided to go to Australia, where he spent seven years, 1962-1968. He was first employed at the Australian National University in Canberra where he stayed until 1964, and then after his discovery of J1 obtained a position of full professor at Monash University in Melbourne, skipping the usual three academic steps. He then moved to the USA, where he spent the period of 1968-1972 first as a visiting professor in Princeton, and then as a full professor at the Ohio State University in Columbus. Since 1972 until his retirement in 2000 Janko was a full professor at the University of Heidelberg, Germany. But even after his retirement his scientific activity is still amazing.

His work can be roughly divided into three parts:

  • theory of finite groups (until 1976),
  • finite geometries and design theory (until 2000), and
  • the theory of p-groups.

Professor Janko is a great fan of theatre and a passionate chess player. In 2007, when he celebrated 75th birthday, his father celebrated 100th brithday in Bjelovar singinig and playing violin.

Here we reproduce a very short, but significant biographical sketch written by professor Zovnimir Janko himself, on the occasion of his very much belated admission to the Croatian Academy of Sciences and Arts in 1993. His admission was not possible ex- Yugoslavia. The biography was written in Croatian language, and published in the Notices of the Croatian Academy of Sciences (Vjesnik Hrvatske akademije znanosti i umjetnosti), see [Janko, p 180].


Biografska bilješka

Zvonimir Janko rođen je 1932. u Bjelovaru, gdje je 1950. završio gimnaziju. Studirao je matematiku na Prirodoslovno-matematičkom fakultetu u Zagrebu od 1950. do 1956. kada je diplomirao. Kao student III godine bio je optužen za hrvatski nacionalizam (u vezi nošenja vijenca na Starčevićev grob), te ga je disciplinski sud kaznio isključenjem s Fakulteta na 2 godine. Godine 1960. doktorirao je i objavio desetak radova u Matematičkom glasniku te u Acta Scientiarum Mathematicorum. Međutim, kad se nakon toga trinaesti puta natjecao za mjesto asistenta ili docenta nigdje u bivšoj Jugoslaviji nije bio primljen, jer ga je stalno pratila negativna politička karakteristika zbog spomenute presude. Stoga je napustio zemlju i od tada mu je akademska karijera sljedeća:

Australian National University, Canberra, 1962-1964 (znanstveni suradnik);

Monash University, Melbourne, Australia, 1965-1968 (redoviti profesor); [u članku pogrešno stoji "znanstveni suradnik"]

Institute for Advance Study, Princeton, N.J., USA, 1968-1969 (gostujući profesor);

Universität Heidlberg od 1972 (redoviti profesor)


Professor Janko delivered a lecture at the Croatian Academy of Sciences and Arts in Croatian, entitled "Kako sam pronašao četiri sporadične grupe" (How I discovered four sporadic groups), March 24th, 1993, published with the above biographical sketch in [Janko].


Source: [Šiftar]


Zvonimir Janko (Bjelovar 1932.), znameniti je matematičar. Osnovnu školu i gimnaziju pohađao je u Bjelovaru. Studirao je matematiku u Zagrebu, gdje je diplomirao (1956.) i doktorirao (1960). Nakon diplome radio je u Širokom Brijegu (tada Lištica) u Bosni i Hercegovini. Razdoblje 1962. - 1968. proveo je u Australiji, najprije u Canberri (1962 - 64.), a zatim kao redoviti profesor na Monash University u Melbourneu (1965.-68.). Od 1968. do 1972. Janko je proveo u SAD kao gostujući profesor na sveučilištu Princeton a potom kao redoviti profesor na Ohio State University u Columbusu (Ohio). Od 1972. do umirovljenja (2000.) redoviti je profesor Sveučilišta u Heidelbergu (Njemačka).

Z. Janko je poznat po dostignuća iz matematičke discipline teorije konačnih grupa. Najznačajnija su mu otkrića novih sporadičnih prostih grupa, danas poznatim pod nazivom Jankove grupe. Ostavio je neizbrisiv trag i u teoriji projektivnih ravnina i dizajna.

Profesor Zvonimir Janko sudjelovao je u poslijediplomskoj nastavi na Matematičkom odjelu Prirodoslovno-matematičkog fakulteta u Zagrebu, održao je niz predavanja u Zagrebu, Splitu i Mostaru. Suradnik je i mentor mnogim hrvatskim i stranim matematičarima. Dopisni je član Hrvatske akademije znanosti i umjetnosti. Tijekom rada u Heidelbergu zapažena je njegova pomoć Hrvatskoj u različitim oblicima.



Professor Zvonimir Janko had 18 students,
who earned ther PhD titles in the fields of group theory and finite geometries.
He has 83 descendants (as of 2022), see his Mathematical Geneaology.

The audience in full concentration...

Professor Goran Muić belongs to younger generation of Croatian top mathematicians.

Monographs dealing with Janko groups

Michael Achbacher, Sporadic Groups, Cambridge Tracts in Mathematics, 104, 1994

Chapter 16. Groups of Conway, Suzuki, and Hall-Janko type

Daniel Gorenstein:

Cheryl E. Praeger, Leonard H. Soicher: Low Rank Representations and Graphs for Sporadic Groups, Australian Mathematical Society Lecture Series, Cambridge University Press, 1997.

The book contains six sections dealing with Janko groups:

  • 4.5 The Janko group J1
  • 4.8 The Janko group J2
  • 4.9 Automorphism group of J2:2 of J2
  • 4.13 The Janko group J3
  • 4.14 Automorphism group of J3:2 of J3
  • 4.35 The Janko group J4

Robert Griess, Twelve Sporadic Groups, Springer monographs in Mathematics, Springer, 1998

Chapter 10: Subgroupsw of Conway Groups; the Simple Groups of Higman-Sims, McLaughlin, Hall-Janko and Suzuki

Book Description
The finite simple groups come in several infinite families (alternating groups and the groups of Lie type) plus 26 sporadic groups. The sporadic groups, discovered between 1861 and 1975, exist because of special combinatorial or arithmetic circumstances. A single theme does not capture them all. Nevertheless, certain themes dominate. The 20 sporadics involved in the Monster, the largest sporadic group, constitute the Happy Family. A leisurely and rigorous study of two of their three generations is the purpose of this book. The level is suitable for graduate students with little background in general finite group theory, established mathematicians and mathematical physicists.

A.A. Ivanov: The Forth Janko Group, Oxford Mathematical Monographs, 2004, 250 pp

Alexandar Antolievic Ivanov is professor of pure mathematics at Imperial College, London, trained as a mathematician in Moscow, Russia

Description of the monograph
This unique reference illustrates how different methods of finite group theory including representation theory, cohomology theory, combinatorial group theory and local analysis are combined to construct one of the last of the sporadic finite simple groups - the fourth Janko Group J_4. Aimed at graduates and researchers in group theory, geometry and algebra, Ivanov's approach is based on analysis of group amalgams and the geometry of the complexes of these amalgams with emphasis on the underlying theory. An indispensable resource, this book will be a unique and essential reference for researchers in the area.

The title of the book on the front conver is simply J4 :

And the inside title reveals to full title of Ivanov's monograph:

Gerhard Michler: Theory of Finite Simple Groups, Cambridge University Press, 2006

Section 8.5: Janko's sporadic groups, pp. 409-418

Chapter 9: Janko group J1, pp 433-454



Professor Vladimir Volenec, behind him professor Goran Muić, dean of the Department of Mathematics of the University of Zagreb, in the next row professor Mario Essert.

Formulation of the Classification Theorem for finite simple groups in Croatian.

Professor Vladimir Ćepulić, delivering an opening lecture on the work of Zvonimir Janko in group theory, and discussing the Classification Theorem for finite simple groups.

Professor Vladimir Ćepulić delivering his lecture about Janko's work in group theory; Janko in the front row. V. Ćepulić is a specialist in group theory, and a PhD student of Dieter Held.

Elizabeta Kovač-Striko, Zvonimir Janko, Vladimir Ćepulić (lecturer), and Zora Janko

Professor Vladimir Ćepulić

Ready for the next lecture to be delivered by...

Professor Juraj Šiftar, University of Zagreb, a student of professor Janko.

Professors Juraj Šiftar and Mario-Osvin Pavčević, two generations of professor Janko's students.

Professor Mario-Osvin Pavčević opening the scientific meeting

Professor Goran Muić, dean of the Department of Mathematics of the University of Zagreb, in his address in honour of professor Zvonimir Janko.


Mark Ronan

Sporadic Groups

In the mid-to-late nineteenth century, the French mathematician Émile Mathieu created five very exceptional groups of permutations, the largest of which is called M24. Mathieu's groups did not fit into the later periodic table, and remained the only exceptions for a hundred years, until the Croatian mathematician, Zvonimir Janko found a new one that he published in 1966. This inspired the search for other sporadic groups, and their discovery is an intriguing story involving a variety of methods: some geometric, some involving patterns exhibiting interesting permutations, and some by analyzing possible cross-sections (called 'involution centralizers' in group theory). These latter cases were very technical, and the construction of the sporadic group was a tricky business, usually involving computer techniques. The Monster - the largest sporadic group - was predicted by the cross-section method, but its size and complicated structure rendered computer methods impractical, and it had to be constructed by hand. There are two main threads that led to the Monster. One was the Leech Lattice and the Conway groups.

Source: The Whole Story by Mark Ronan

Invariant Society, The Student Mathematical Society of the University of Oxford

Prof. Mark Ronan, University of Illinois, Visiting Professor at UCL
Symmetry and the Monster (lecture delivered in 2006)

This talk will describe the quest to find a complete list of all finite simple groups. These groups, the basic building blocks for all finite groups, were instrumental in Galois' work on the solvability of algebraic equations; and he himself discovered some important ones. After the Second World War new families of simple groups were discovered, and there was enormous interest in finding a complete list. Whilst this was going on, however, Zvonimir Janko, a Croatian mathematician working in Australia, surprised the world with a very strange exceptional group. This was the first exception since Émile Mathieu discovered five beautiful groups of permutations in the nineteenth century, and it really set the cat among the pigeons. Further new exceptions came thick and fast, and they were dubbed "sporadic groups". The largest is called the Monster. This talk will explain how the Monster was discovered, and how it came to reveal strange connections between number theory and mathematical physics, sometimes called the moonshine connections.


Professor Mario-Osvin Pavčević, University of Zagreb, one of the organizers of the meeting on the occasion of 75th birthday of professro Janko.

Professor Zvonimir Janko in friendly discussion with two generations of his indirect students: Professor Vladimir Ćepulić (University of Zagreb) and dr. Sanja Rukavina (University of Rijeka).

Some of the activities related to the work of Zvonimir Janko

The number n of involutions in a finite 2-group G is odd. If n=1, then it is known for already 200 years (and available in any textbook on algebra) that then the group G is either cyclic or a generalised quaternionic group. However, the case of of n=3 is surprisingly difficult, but exceptionally important in the theory of finite groups. Despite some attempts to solve this problem (A.D.Ustjuzaninov in 1972 and M. Konviser 1973), it was only in 2004 that due to results of Zvonimir Janko the 2-groups G with n=3 have been satisfactorily classified.

Professor Janko delivered a lecture in November 11th, 2007 about his results at the Scientific Colloquium of the Croatian Mathematical Society, entitled: Rjesenje problema tri involucije (A solution of the problem of three involutions).

Broj involucija n u konačnoj 2-grupi G je neparan. Ako je n=1, tada je poznato već 200 godina (i nalazi se u svakom udžbeniku algebre) da je grupa G ciklička ili generalizirana kvaternionska grupa. Međutim, sljedeći slučaj n=3 je začuđujuće težak, ali izrazito važan za teoriju konačnih grupa. Mada je bilo pokušaja, da se taj problem riješi (A.D.Ustjuzaninov 1972. i M.Konviser 1973.), tek su ovogodišnjim rezultatima Zvonimira Janka 2-grupe G s n=3 klasificirane na zadovoljavajući način.

In May 26, 2004, at the Colloquium of the Rudjer Boskovic Institute in Zagreb, professor Janko delivered a lecture entitled Moderna teorija konačnih p-grupa (Modern theory of finite p-groups).

February 14, 2005, Colloquium at the Freie Universität Berlin, Germany

Barbara Baumeister -Freie Universität Berlin

The third group of Janko

Abstract: In an enormous proof all the finite simple groups have been classified. (Recall that a group is simple if it has no normal subgroup other than 1 and G.) There are some infinite families and 26 sporadic exceptions. One of them is specially difficult to handle, the third group of Janko, J3. Janko gave evidence that there is a simple group G satisfying the two conditions:

  • If i and j are involutions in G (i.e. i2 = j2 =1), then there is g in G such that g-1 i g = j.
  • Let i be an involution in G and CG(i) = {g in G | ig = gi}, which is a subgroup of G. Then CG(i) is a prescribed group with 27 x 3 x 5 elements.

Such a group is called "group of J3-type". He also showed that a group of J3-type has 50.232.960 elements. In my PhD thesis I proved the existence of a group of J3-type by constructing a geometry with automorphism group isomorphic to J3. Using this information I am now trying to show that there is just one group of J3-type up to isomorphism.


13th British Combinatorial Conference, University of Surrey, Guildford, 8-12 July 1991

... one of contributed talks: M. Weidenfeld, A construction for the graph associated with the small Janko group

Algebra Research Group, Oxford

November 20, 2001, Dr. M. Holloway Oxford,
Broue's conjecture, the Hall-Janko group and computer calculations

Miles Holloway: Broue's conjecture for the Hall-Janko group and its double cover. Proc. London Math. Soc. (3), 86(1):109-130, 2003.

Université Libre de Bruxelles, Département de Mathématique

Service de Géometrie

1996, D. Leemans:
Study of the residually weakly primitive flag-transitive geometries of the first Janko group , the Mathieu group and the Suzuki groups. Study of an Ivanov-Shpectorov geometry for the O'Nan sporadic group.

Department of Mathematical Sciences

Binghamton University, State University of New York

The Algebra Seminar, 2007

October 16: Hyun Kyu Kim (Cornell)
Title: Simultaneous construction of the sporadic groups Co_1, Fi_24' and J_4
Abstract: Recently G. Michler formulated an algorithm constructing certain finite simple groups G from well-defined extensions E of irreducible subgroups T of GL_n(2) by an n-dimensional FT-module A, where F is the field with 2 elements. In this lecture I apply this algorithm in 3 cases where A is an irreducible FT-module of dimension n=11 and T is the Mathieu group M_24. The algorithm allows us to construct Conway's sporadic group Co_1 and Janko's sporadic group J_4 inside GL_276(23) and GL_1333(29), respectively (and hopefully Fischer's sporadic group Fi_24' inside GL_8671(13)).

University of Birmingham, School of Mathematics

Pure Maths Research Staff & Activities, 2007

Finite Classical and Sporadic Groups (Professor Rob Curtis, Professor Chris Parker, Professor Sergey Shpectorov, Dr Corneliu Hoffman, Dr Yongzhong Sun, Emeritus Professor J Wilson)

... Curtis has provided fresh insight into the Mathieu groups and done a lot to explain the existence of other sporadic groups. His approach yields presentations and constructions of these groups which reflect their geometry and symmetry and which are easy to work with. His techniques have recently yielded an elementary proof of the existence of the smallest Janko group. Additionally, the largest Janko group and the Conway group .O have been constructed from this point of view. This approach also reveals remarkably analogous constructions for the Held group and the Harada-Norton groups, which are respectively centralizers of an element of order 7 and an element of order 5 in the Monster. One of his goals is to find a uniform method for the construction of these exceptional and remarkable structures and together with Parker he has had funding from the EPSRC for this purpose. ...

Representation Theory, General Finite Groups and Associated Geometries (Professor R Curtis, Professor CW Parker, Dr PJ Flavell, Dr AD Gardiner, Dr Corneliu Hoffman, Dr I Korchagina)

... Professor Rob Curtis, in collaboration with Dr John Bray, has produced an effective double coset enumeration procedure for symmetrically presented groups; this program has been implemented by the latter researcher and works well for large groups, including the largest Janko group J4.

The classification of the finite simple groups requires about 15,000 journal pages. Many mathematicians world-wide are trying to give new and better proof for results needed in the classification. Dr Inna Korchagina works jointly with Professors Ron Solomon and Richard Lyons, the two people most heavily involved in producing the definitive classification of finite simple groups after the untimely death of Daniel Gorenstein. Her recent work characterizes simple groups such as the Suzuki and Thompson groups which have mixed characteristics.

Robert Curtis: Symmetric Generation of Groups: With Applications to many of the Sporadic Finite Simple Groups (Encyclopedia of Mathematics and its Applications), Cambridge University Press, 2007, 332 pp

Book Description
Some of the most beautiful mathematical objects found in the last forty years are the sporadic simple groups. However, gaining familiarity with these groups presents problems for two reasons. First, they were discovered in many different ways, so to understand their constructions in depth one needs to study lots of different techniques. Second, since each of them is in a sense recording some exceptional symmetry in spaces of certain dimensions, they are by their nature highly complicated objects with a rich underlying combinatorial structure. Motivated by initial results which showed that the Mathieu groups can be generated by highly symmetrical sets of elements, which themselves have a natural geometric definition, the author develops from scratch the notion of symmetric generation. He exploits this technique by using it to define and construct many of the sporadic simple groups including all the Janko groups and the Higman-Sims group. This volume is suitable for researchers and postgraduates.

About the Author
Robert T. Curtis is Professor of Combinatorial Algebra and Head of Pure Mathematics in the School of Mathematics at the University of Birmingham. He also holds the post of Librarian for the London Mathematical Society.


Professor Zvonimir Janko with professor Elizabeta Kovač-Striko, University of Zagreb, on the right, also his PhD his student.

Dr. Tanja Vučičić and Dr. Snježana Braić, students of professor Zvonimiro Janko from the University of Split.

Finite Geometries and Design Theory

Professor Zvonimir Janko worked in this fields in the period of 1980-2000. Among his close collaborators we mention Tran van Trung, Vladimir Tonchev, and Hadi Kharagani. In 1987 he delivered a minicourse at the Mathematics Department in Zagreb, consisting of five lectures under the common title Design Theory.

Finite Geometry

Groups of Prime Power Order

Yakov Berkovich, a Russian mathematician now working at the University of Haifa, Israel (address: Department of Mathematics, University of Haifa, Mount Carmel, Haifa 31905, Israel), has a very fruitful collaboration with Zvonimir Janko in the past years. He is preparing a monograph entitled "Groups of Prime Power Order", which was originally planned to be issued in 2001. However, Zvonimir Janko has obtained so many substantially new results that Berkovich postponed issuing the book, since otherwise the book would be immedately outdated. Moreover, professor Berkovich claimed in 2001 that some of the results that Janko had recently obtained in this field are the most important in the past 30 years! Meanwhile, the material has grown enormously, and now the monograph will be issued in three parts, starting with 2008. The second and third part will be a joint work of professors Berkovich and Janko.

Saunders MacLane placed Zvonimir Janko among originators of the theory of p-groups.

A voluminous hardcover monogrpah issued in 2008, 520 pp.

Vol 2, Vol 3, Vol 4, Vol 5, and

Berkovich, Yakov G.; Janko, Zvonimir Groups of prime power order. Vol. 6. De Gruyter Expositions in Mathematics, 65. De Gruyter, Berlin, 2018. xxi+385 pp.

For more information see MathSciNet.


Professor Zvonimir Janko at the University of Zagreb, 2007

A lecture by professor Zvonimir Janko in Zagreb in June 2017

On June 2nd, 2017, Professor Zvonimir Janko visited the Faculty of Electrical Engineering and Computing of the University of Zagreb in order to delivered a lecture entitled "Čudno ponašanje konačnih p-grupa" (in Croatian, Strange behavior of p-groups), organized within the Seminar for finite geometries and groups. The seminar has been opened by one of its leaders, Professor Mario Osvin Pavčević. The talk has raised a great interest, so that some colleagues from the Universities of Split and Rijeka arrived to Zagreb especially for that occasion.

During the talk, Professor Janko mentioned among others that it is expected that the 6th (and the last) volume of the monograph "Groups of Prime Power Order" written (since volume 2) jointly with Professor Yakov Berkovich, Israel, contains as many as 4000 open problems. Out of them, about 1000 he considers to be tractable, another 1000 as untractable (sic!), and for the rest (2000 problems) he believes that each of them can serve as an individual theme for a PhD dissertation. He stressed that the books have raised a great interest, especially in China.

Berkovich, Yakov G.; Janko, Zvonimir Groups of prime power order. Vol. 6. De Gruyter Expositions in Mathematics, 65. De Gruyter, Berlin, 2018. xxi+385 pp.

For more information see MathSciNet.

Professor Zvonimir Janko on June 2nd, 2017, as a guest of the University of Zagreb.

Professor Zvonimir Janko with Professors Tanja Vučičić and Anka Golemac,
his colleages from the University of Split.

Professor Zvonimir Janko, Univeristy of Heidelberg, with Professors Ljubo Marangunić
and Zijad Haznadar, University of Zagreb.

Professor Mario Osvin Pavčević opening opening the seminar with the talk of
Academician Zvonimir Janko, organized at the University of Zagreb on June 2nd, 2017.

Profeesor Zvonimir Janko starting his talk "Čudno ponašanje konačnih p-grupa"
(in Croatian, Strange behavior of p-groups)

Nine photos that follow have been provided by Professor Ljiljana Bralić from the University of Split:

Zvonimir Janko with Anka Golemac, professor at the University of Split

Professor Zvonimir Janko is an agreable and communicative person.

Answering questions

Jankova otkrića potresla su matematičke znanstvenike u svijetu (interview, in Croatian)

Theorem 231.1, formulated in Croatian

Professors Snježana Braić of the University of Split and Sanja Rukavina of the University of Rijeka.

Professors Anka Golemac, Zvonimir Janko and Tanja Vučičić.

Many thanks to Professor Ljiljana Bralić, University of Split, for providing the above 9 photos.

Collaboration with Croatian mathematicians

In winter semester of 2000 Janko taught a postgraduate course Finite Groups Theory (30 hours) at the University of Zagreb, Department of Mathematics, PMF.

In the years of 2000-2003 professor Janko delivered courses for graduate students at the Department of Mathematics, University of Zagreb, in which he introduced listeners into the field of his current research - the theory of finite p-gourps. The lectures were delivered in his characteristic way - clearly and precisely, and at the same time with a lots of humour. He taught from the very beginning, lecturing elements of group theory, till contemporary probelms of current investigations. See [Ćepulić].

He had numerous lectures in group theory and design theory, and his students work in Zagreb, Rijeka and Split (in Croatia), and in Mostar (in Bosnia and Herzegovina). He supervised many PhD theses in Croatia.

Dr. Zdravka Božikov, University of Split, was among the first Croatian mathematicians to earn her PhD thesis (1984) under the guidance of professor Janko. Zvonimir Janko has many collaborators, not only in Croatia, so that one can speak about Janko's School, see [Šiftar].

In 2007 professor Janko obtained a special recognition from two German institutions, German Rector's Conference and German Research Foundation (Deutsche Forschungsgemeinschaft, DFG), for his contribution to the development of mathematics research in South East Europe. His students are in Croatia, Bosnia and Herzegovina, and Kosovo, see [Ćepulić].


Mathematical notions bearing the name of Janko

  • Janko groups
  • Janko's sporadic groups
  • the first group of Janko
  • the second group of Janko
  • the third group of Janko
  • the fourth group of Janko
  • the largest Janko group
  • the smallest Janko group
  • Janko's simple group
  • Hall-Janko group
  • Hall-Janko graph
  • Higman-Janko-McKay group

References related to Janko groups

Source: Ben Fairbairn, University of Birmingham, Sporadic Group References

Janko group J1

  • P.J.Cameron "Another characterization of the small Janko group " (Kovas and Neuman, eds.), pp 205-208, Gordan and Breach, London and New York, 1967
  • G.R.Chapman, "Generators and relations for the cohomologyring of Janko's first group in the first twenty-one dimensions", in 'Groups - St. Andrews 1981' (Campbell and Robertson eds.) CUP, 1982
  • J.H.Conway "Three lectures of exceptional groups", J.H.Conway and N.J.A.Sloane "Sphere Packings, Lattices and Groups", Springer-Verlag New York Inc., 2003
  • R.T.Curtis and Z.Hassan "Symmetric representation of elements of elements of the Janko group J1", J. Symbolic Comput. 22 (1996), 201-214
  • R.T.Curtis "Symmetric generation and existence of the Janko group J1" J. Group Theory 2 (1999) 355-366
  • P.Fong "On the decomposition numbers of J1 and R(q)", Symp. Math. 13 (1972) 415-422
  • T.M.Gagen "On groups with Abelian Sylow 2-Sylow subgroups", in 'Theory of Finite Groups' (Kovacs and Neuman, eds.), pp. 99-100, Gordon and Breech, London and New York, 1967
  • T.M.Gagen "A characterization of Janko's simple group", Proc.Amer. Math. Soc. 19 (1968) 1393-1395
  • G.Higman "Constructions of simple groups from character tables", in 'Finite Simple Groups'(Powell and Higman, eds.) pp. 205-214, Academic Press, 1971
  • Z.Janko "A new finite simple group with Abelian Sylow 2-subgroups", Proc. Nat. Acad. Sci. USA 53 (1965) 657-658
  • Z.Janko "A new finite simple group with Abelian Sylow 2-subgroups and its characterization", J. Algebra 3 (1966) 147-186
  • Z.Janko "A characterization of a new simple group", in 'The Theory of Groups' (Kovac and Neumann, eds.), pp.205-208,Gordon and Breach,London and New York, 1967
  • P.Landrock and G.O.Michler, "Block structure of the smallest Janko group", Math Ann. 232 (1979) 205-238
  • C.S.Li "The commutators of the small Janko group J1", J. Math (Wuhan) 1 (198) 175-179
  • D.Livingstone "On a permutation representation of the Janko group", J. Algebra 6 (1967) 43-55
  • R.P.Martineau "A characterization of Janko's simple group fo order 175,560" Proc. Lon. Math. Soc. 19 (1969) 709-729
  • M.Perkel "A characterization of J1 in terms of its geometry", Geom. Ded. 9 (1980) 291-298
  • E.E.Shult "A note on Janko's group of order 175,560", Proc. Amer. Math. Soc. 35 (1972) 3442-348
  • T.A.Whitelaw "Janko's group as a colineation group in PG(6,10), Proc. Cambridge Philos. Soc. 63 (1967) 663-677
  • H.Yamaki "On Janko's simple group of order 175,560", Osaka J. Math. 9 (1972) 111-112
  • R.A.Wilson "Is J1 a subgroup of the Monster?" Bull. London. Math. Soc. 18 (1986), 349-350.

Hall-Janko group J2/HJ

  • A.M.Cohen "Finite quarternionic reflection groups", J. Algebra (1980) 293-324
  • L.Finkelstein and A.Rudvalis "Maximal subgroups of the Hall-Janko-Wales group", J. Algebra 24 (1973) 486-493
  • S.M.Gagola and S.C.Garrison "Real charaters, douls covers and the multiplier", J. Algebra 74 (1982) 20-51
  • D.Gorenstein and K.Harada "A characterisation of Janko's two new simple groups",J. Fac. Sci. Univ. Tokyo 16 (1970) 331-406
  • M. Hall and D.S.Wales "The simple group of order 604,800" J. Algebra (1968) 417-450
  • A.P.Il'inyh "A characterisation of the Hall-Janko finite simple group", Mat. Zametki 14 (1973) 535-542
  • Z.Janko "Some new simple groups of finite order", in 'Theory of finte groups' (Brauer and Sah, eds.), pp. 97-100, Benjamin, 1969
  • J.H.Lindsey "On a projective representation of the Hall-Janko group", Bull. Amer. Math. Soc. 74 (1968) 1094
  • J.H.Lindsey "Linear groups of degree6 and the Hall-Janko group", in 'The Theory of Finite Groups' (Brauer and Sah, eds.), pp. 97-100, Benjamin, 1969
  • J.H.Lindsey "On the 6-dimensional projective representation of the Hall-Janko group, Pacific J. of Math. 35 (1970) 175-186
  • J.McKay and D.B.Wales "The multipliers of the simple groups of orders 604,800 and 50,232,960", J. Algebra 17 (1971) 262-272
  • F.L.Smith "Ageneral characterisation of the Janko simple group J2", Arch. Math. (Basel) 25 (1974) 17-22
  • J.Tits "Le groupe de Janko d'order 604,800", in 'The Theory of Finite Groups' (Brauer and Sah, eds.), pp. 97-100, Benjamin, 1969
  • D.B.Wales "The uniquness of the simple group of order 604,800 as a subgroup of SL6(4)", J. Algebra 11 (1969) 455-460
  • D.B.Wales "Generators of the Hall-Janko group as a subgroup of G2(4), J. Algebra 13 (1969) 513-516
  • R.A.Wilson "The geometry of the Hall-Janko group as a quarternionic reflection group", Preprint, Cambridge (1984)

Janko group J3

  • M.Aschbacher "The existance of J3 and its embeddings in E6" Geom. Dedicata 35 (1-3) (1990) 143-154
  • J.D.Bradley "Symmetric Presentations of Two Sporadic Groups ", Ph.D. thesis, Birmingham (2005)
  • J.H.Conway and D.B.Wales "Matrix generators for J3", J.Algebra 29 (1974) 474-476
  • L.Finkelstein and A.Rudvalis "The maximal subgroups of Janko's simple group of order 50,232,960", J.Algebra 30 (1974) 122-143
  • D.Frohardt "A trilinear form for the third Janko group", J. Algebra 83 (1983) 349-379
  • D.Gorenstein and K.Harada "A characterisation of Janko's two new simple groups",J. Fac. Sci. Univ. Tokyo 16 (1970) 331-406
  • G.Higman and J.McKay "On Janko's simple group of order 50,232,960", Bull. London Math. Soc. 1 (1969) 89-94
  • Z.Janko "Some new simple groups of finite order", in 'Theory of finte groups' (Brauer and Sah, eds.), pp. 97-100, Benjamin, 1969
  • P.B.Kleidman and R.A.Wilson "J3 < E6(4) and M12 < E6(5)", J. London Math. Soc. (2)42 (1990), 555-561
  • D Leemans "The Residually Weakly Primitive Geometries of J3" Experimental Math. 13 (2004) 429-434
  • J.McKay and D.B.Wales "The multipliers of the simple groups of orders 604,800 and 50,232,960", J. Algebra 17 (1971) 262-272
  • I.A.I.Suleiman and R.A.Wilson "Standard generators J3", Experimental Math 4 (1995) 11-18
  • R.M.Weiss "A geometric construction of Janko's group J3", Math. Z. 179 (1982) 410-419
  • R.M.Weiss "On the geometry of Janko's group J3", Arch. Math. (Basel) 38 (1982) 410-419
  • R.Weiss "A Characterization and Another Construction of Janko's Group J3", Trans. Amer. Math. Soc. 298, (1986) 621-633
  • R.A.Wilson "The 2- and 3-modular characters of J3, its covering group and automorphism group", J.Symbol. Comp. 10 (1990), 647-656
  • R.A.Wilson "The Brauer tree for J3 in characteristic 17", J. Symbol Comp. 15 (1993) 325-330
  • S.K.Wong "On a new finite non-Abelian simple group fo Janko", Bull. Austral. Math. Soc. 1 (1969) 59-79

Janko group J4

  • M.Aschbacher and Y.Segev "The uniquness of groups of type J4" Invent. Math. 105 (3) (1991) 589-607
  • S.W. Bolt, J.N. Bray and R.T. Curtis "Symmetric generation of the Janko group J4", prprint, Birmingham, 2005
  • D.J.Benson "The simple group J4", Ph.D. thesis, Cambridge, (1980)
  • I.S.Guloglu "A characterization of the simple group J4", Osaka J. Math. 18 (1981) 13-24
  • A.A.Ivanov "J4", Clarendon Press (2004)
  • Z.Janko "A new finite simple group of order 86,775,570,046,077,562,880, which posses M24 and the full covering group of M22 as subgroups", J. Algebra 42 (1976) 564-596
  • Jianbei An, E.A.O'Brien and R.A.Wilson "The Alperin weight conjecture and Dade's conjecture for the simple group J4", London Math Soc J Comput Math, to appear, 2003
  • W.Lempken "The Schur multiplier of J4 is trivial", Arch. Math. 30 (1978) 267-270
  • W.Lempken "A 2-local characterization of Janko's simple group J4", J. Algebra 55 (1978) 403-445
  • G.Mason "Some remarks on groups of type J4", Arch. Math. 29 (1977) 574-582
  • S.P.Norton "The construction of J4", in 'The Santa Cruz Confrence on Finite Groups' (Cooperman and Mason, eds.), pp.271-278, Amer. Math. Soc. (1980)
  • A.Reifart "Some simple groups related to M24", J. Algebra 45 (1977) 199-209
  • A.Reifart "Another characterization of Janko's new simple group J4", J. Algebra 49 (1977) 621-627
  • A.Reifart "A 2-local characterization of the simple groups M24, Co1 and J4", J. Algebra 50 (1978) 213-227
  • R.M.Stamford "A characterization of Janko's new simple group J4", Notices Amer. Math. Soc. 25 (1978) A-423
  • R.M.Stamford "A characterization of Janko's new simple group J4 by centralizers of elements of order three", J. Algebra 57 (1979) 555-566
  • G.Stroth "An odd characterization of J4", Israel J. Math. 31 (1978) 189-192
  • G.Stroth and R.Weiss "Modified Steinberg Relations for the Group J4", Geom. Dedicata 25, (1988) 513-525
  • P.B.Kleidman and R.A.Wilson "The maximal subgroups of J4", Proc. London Math. Soc. (3)56 (1988) 484-510




  • Zvonimir Janko at MathSciNet (containing his 126 his scientific publications)
  • Michael Aschbacher, Sporadic Groups, Cambridge Tracts in Mathematics, 104, 1994
  • Michael Aschbacher, The Status of the Classification of the Finite Simple Groups [PDF], Notices of the American Mathematical Society, August 2004
  • Y. Berkovich and Z. Janko, Structure of finite p-groups with given subgroups, in Contemp. Math. 402, Amer. Math. Soc., Providence, 2006, 13-93.
  • Y. Berkovich, Groups of Prime Power Order, Part I, De Gruyter, 2008
  • Y. Berkovich and Z. Janko, Groups of Prime Power Order, Part II, De Gruyter, 2008, 520 pp, more information
  • Y. Berkovich and Z. Janko, Groups of Prime Power Order, Part III, De Gruyter
  • Y. Berkovich and Z. Janko, Groups of Prime Power Order, Parts IV, V, and
  • Berkovich, Yakov G.; Janko, Zvonimir Groups of prime power order. Vol. 6. De Gruyter Expositions in Mathematics, 65. De Gruyter, Berlin, 2018. xxi+385 pp.,
    For more information see MathSciNet.
  • Vladimir Ćepulić, Profesor Zvonimir Janko i teorija grupa, prikaz na proslavi u čast profesora Janka 26. listopada 2007. u godini njegovoga 75. rođendana (in Croatian)
  • Vladimir Devidé: Znam da ću morati otići i ne vratiti se ..., Marulic, br. 6, Zagreb 2008., 1022-1032
  • Ben Fairbairn, Sporadic Group References
  • James Franklin, Non-deductive Logic in Mathematics, British Journal for the Philosophy of Science 38 (1987), pp. 1-18
  • Daniel Gorenstein, Finite Simple Groups, Plenum Press, 1982
  • Daniel Gorenstein, Richard Lyons, Ronald Solomon, The Classification of the Finite Simple Groups (Volume 1 - available online!), AMS, 1994 (Volume 2 - available online), AMS, 1996
  • H. Gottschalk, D. Leemans, The residually weakly primitive geometries of the Janko group J1, in: A. Pasini et al. (Eds.), Groups and Geometries, Birkhäuser, 1998, pp. 65-79.
  • Robert Griess, Twelve Sporadic Groups, Springer monographs in Mathematics, Springer, 1998
  • Groupe de Janko, at fr.wikipedia.org
  • Zeljko Hanjs: Interview with professor Zvonimir Janko (in Croatian), Matematicko-fizicki list, 2009/2010 no 1, pp. 10-11. (in this interview professor Janko said that he calls J1 also Zora's group in honour to his wife)
  • Michael Hartley: Polytopes Derived from Sporadic Simple Groups - Auxiliary Information; The First Janko Group J1 (300 polytopes), The Second Janko Group J2 (292 polytopes), The Third Janko Group J3 (586 polytopes)
  • Dieter Held: Die Klassifikation der endlichen einfachen Gruppen [PDF], survey article
  • Marc Hindry (Paris VII), List des groupes simples finis [PDF]
  • Naoyuki Horiguchi, Masaaki Kitazume, Hiroyuki Nakasora, The Hall-Janko graph and the Witt system W10, European Journal of Combinatorics,
    Volume 29 , Issue 1 (January 2008), 1-8
  • A.A. Ivanov (professor at Imperial College, London): The Fourth Janko Group (256 pp), Oxford Math. Monographs, 2005.
  • Zvonimir Janko, A new finite simple group with abelian 2-Sylow subgroups, Proc. National Academy of Sciences USA, 53, 657-658 [announcement of the main results in the following paper, regarding J1]
  • Zvonimir Janko, A new finite simple group with abelian Sylow subgroups and its characterization, Journal of Algebra 3: 147-186, (1966) doi:10.1016/0021-8693(66)90010-X [the group was subsequently named J1]
  • Zvonimir Janko, Some new finite simple groups of finite order, 1969 Symposia Mathematica (INDAM, Rome, 1967/68), Vol. 1 pp. 25-64 Academic Press, London MR0244371 [the new groups were subsequently named J2 and J3], see also The theory of finite groups (Editied by Brauer and Sah) p. 63-64, Benjamin, 1969.
  • Zvonimir Janko, A new finite simple group of order 86,775,570,046,077,562,880 which possesses M24 and the full covering group of M22 as subgroups, J. Algebra 42 (1976) 564-596. doi:10.1016/0021-8693(76)90115-0 [the group was subsequently named J4]
  • Janko, Zvonimir. Kako sam pronašao četiri sporadične grupe / Zvonimir Janko. // Vjesnik Hrvatske akademije znanosti i umjetnosti. God. 2(1993), br. 1/3 / glavni urednik Vladimir Paar
  • Janko, Zvonimir, in Hrvatski biografski leksikon (Croatian Biographical Lexicon), 6, I-Kal, pp 322-323
  • Janko, Zvonimir, references at MathSciNet
  • Janko Group
  • J1, J2, J3, J4 at ATLAS of finite group representations
  • Alexander Konovalov (Olexandr Konovalov), Janko groups in the area of group rings
  • Mathematics Geneaology: Zvonimir Janko
  • Gerhard O. Michler and Michael Weller: A new computer construction
    of the irreducible 112-dimensional 2-modular representation of Jankoćs
    group J4. Comm. Algebra, 29(4):1773-1806, 2001.
  • Gerhard Michler: Theory of Finite Simple Groups, Cambridge University Press, 2006
  • Orders of nonabelian simple groups (includes a list of all non-abelian simple groups up to order 10,000,000,000)
  • Marc Ronan: Symmetry and the Monster, One of the Greatest Quests of Mathematics, Oxford University Press, 2006 (mathematicians involved in this book, including professor Janko); book review [PDF] by Robert L. Griess in the Notices of the American Mathematical Society, February 2002
  • Juraj Šiftar: Profesor Zvonimir Janko (in Croatian and English), in Ugledni hrvatski znanstvenici u svijetu I. (Distinguished Coratian Scientists in the World, ed. Janko Herak), Hrvatsko-Američko društvo i Hrvatska matica iseljenika, Zagreb 2002, pp 9-27, Sažetak
  • Ron Solomon: On Finite Simple Groups and their Classification [PDF], Notices of the American Mathematical Society, February 1995
  • Ronald Solomon: A brief history of the classification of the finite simple groups, Bull. Amer. Math. Soc. 38 (2001), pp. 315-352.
  • Sporadic groups
  • The Whole Story by Mark Ronan
  • Tits, J., Le groupe de Janko d'ordre 604, 800. In: Theory of Finite Groups (Symposium, Harvard Univ, Cambridge, Mass, 1968), Benjamin, New York. pp. 91-95.


Lectures (videos)

Robert Griess: My life and times with the sporadic simple groups

John Conway: The Monster Group

Monster Group (John Conway) - Numberphile

Monster Group (a little extra bit)

John Thompson - Speaking at Finite Simple Groups: Thirty Years of the Atlas and Beyond

Richard E. Borcherds:  Sporadic groups

Richard E. Borcherds: Sporadic groups and number theory

Richard Borcherds: Sporadic Groups - The Archimedeans

Peter. E. Neumann: A breakthrough in Algebra: Classification of the Finite Simple Groups - LMS 1992

Simple Groups - Abstract Algebra

Group theory, abstraction, and the 196,883-dimensional monster

10,000 page proof! The Classification of Finite Simple Groups

Simple groups, Lie groups, and the search for symmetry II | Math History | NJ Wildberger


Acknowledgements. Many thanks to professor Zvonimir Janko for kindly providing his 1964 photo upon my request. I express my deep gratitude to professor Vladimir Ćepulić, University of Zagreb, for his illuminating article dealing with life and work of professor Zvonimir Janko. Photos from the 2007 conference held in honour of professor Janko which are of the form janko_m*.jpg are shot by professor Mario Pavčević, janko_k*.jpg are by Kristijan Tabak, M.Sc, janko_s*.jpg by Siniša Miličić, dipl.ing., and the rest is by Darko Žubrinić, the author of this web page.

William Feller, distinguished Croatian - American mathematician

Alex Grossmann, distinguished Croatian-French physicist and mathematician,
a founder of Wavelet Theory

Croatian Science