Breakthrough in the modern theory of sporadic groups
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 . 
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:  M_{11}, M_{12}, M_{22}, M_{23}, M_{24}
 J_{1}, J_{2}, J_{3}, J_{4}
 Hs
 Co_{1}, Co_{2}, Co_{3}
 He
 Mc
 Suz
 M(22), M(23), M(24)'
 Ly
 Ru
 ON
 F_{5}, F_{3}, F_{2}, F_{1}
 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 30year 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 groupsat the International Congress of Mathematicians in Nice, France, in 1966. 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 (18351890). Mathieu goups M_{11} and M_{12} have been discovered in 1861, and M_{22}, M_{23}, and M_{24} 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 J_{1} in his honour. It has 175,650 elements. The discovery of J_{1} 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:  J_{2} (with Hall, also denoted HJ and called HallJanko group), in 1966,
 J_{3} (with G. Higman and McKay, also called HigmanJankoMcKay group, HJM), also in 1966, and
 J_{4} (with Norton and Parker) in 1975.
In 1967 Higman and Sims constructed sporadic group HS. John Conway constructed three sporadic groups, Co_{1} and Co_{2} in 1968, and Co_{3} in 1968. Dieter Held, a student of Professor Janko from Mainz, 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: F_{5} by John Thompson and Stephen Smith, F_{3} by Koichiro Harada, Simon Norton and Scmith, and F_{1} by Bernd Fischer and Robert Griess. The sporadic group F_{1} is the largest one, better known as the Monster group. It has approximately 8x10^{53} elements, or precisely 808,017,424,794,512,875,886,459,904,961,710,757,005,754,368,000,000,000 elements. The second largest one is F_{2}, called the baby Monster, which has 4,154,781,481,226,426,191,177,580,544,000,000 elements. The fourth largest sporadic group is J_{2}, with order equal to 86,775,571,046,077,562,880 It should be stressed that Zvonimir Janko not only discovered the first sporadic group in the 20th century (J_{1} in 1964), but also the last one (J_{4} 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 (M. Suzuki, D. Held, I. Conway, I. Thompson, I. Sims, Rudvalis, R. Lyons, Harada and Bernd Fischer, who discovered the largest sporadic group, the so called Monster). In 1981 it has been proved by S. Norton, Cambridge, 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 quasithin 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 J_{1}. To illustrate the "earthquake" in group theory which raised Janko's discovery of J_{1} we cite the following testimony from [Æepuliæ] (translated from Croatian by D.®.). In 1997, during the celebration of 65th birthday of professor Janko in Maniz, 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 Æepuliæ 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 visible 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!" ([Æepuliæ], translated form Croatian by D.®.)
 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).
SIMPLE GROUPS (Sung to the tune of "Sweet Betsy from Pike") 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 A_{n} 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 theyve'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 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 determinded 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), 609614.) Remark 2. The ballad was written before the discovery of the last sporadic group in 1975, that is, of the Janko group J_{4}. In other words, the ballad is far from being finished.
 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 Li¹tica (also known as ©iroki 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 Li¹tica. 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 pro¹irenja (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 he alreday had about ten published journal articles, satisfying condition 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 Starèeviæ, 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, but in vain (even in Tuzla in Bosnia and Herzgovina.) Therefore Janko decided to go to Australia, where he spent seven years, 19621968. He was firstf employed at the Australian National University in Canberra where he stayed until 1964, and then obtained a position of full professor at Monash University in Melbourne. He then moved to the USA, where he spent the period of 19681972 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 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 pgroups.
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 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 bilje¹ka Zvonimir Janko roðen je 1932. u Bjelovaru, gdje je 1950. zavr¹io gimnaziju. Studirao je matematiku na Prirodoslovnomatematièkom fakultetu u Zagrebu od 1950. do 1956. kada je diplomirao. Kao student III godine bio je optu¾en za hrvatski nacionalizam (u vezi no¹enja vijenca na Starèeviæev grob), te ga je disciplinski sud kaznio iskljuèenjem s Fakulteta na 2 godine. Godine 1960. doktorirao je i objavio desetak radova u Matematièkom glasniku te u Acta Scientiarum Mathematicorum. Meðutim, kad se nakon toga trinaesti puta natjecao za mjesto asistenta ili docenta nigdje u biv¹oj Jugoslaviji nije bio primljen, jer ga je stalno pratila negativna politièka karakteristika zbog spomenute presude. Stoga je napustio zemlju i od tada mu je akademska karijera sljedeæa: Australian National University, Canberra, 19621964 (znanstveni suradnik); Monash University, Melbourne, Australia, 19651968 (redoviti profesor); [u èlanku pogre¹no stoji "znanstveni suradnik"] Institute for Advance Study, Princeton, N.J., USA, 19681969 (gostujuæi 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 prona¹ao èetiri sporadiène grupe" (How I discovered four sporadic groups), March 24th, 1993, published with the above biographic sketch in [Janko]. Source: [©iftar] Sa¾etak Zvonimir Janko (Bjelovar 1932.), znameniti je matematièar. Osnovnu ¹kolu i gimnaziju pohaðao je u Bjelovaru. Studirao je matematiku u Zagrebu, gdje je diplomirao (1956.) i doktorirao (1960). Nakon diplome radio je u ©irokom Brijegu (tada Li¹tica) 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 gostujuæi profesor na sveuèili¹tu Princeton a potom kao redoviti profesor na Ohio State University u Columbusu (Ohio). Od 1972. do umirovljenja (2000.) redoviti je profesor Sveuèili¹ta u Heidelbergu (Njemaèka). Z. Janko je poznat po dostignuæa iz matematièke discipline teorije konaènih grupa. Najznaèajnija su mu otkriæa novih sporadiènih 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 Matematièkom odjelu Prirodoslovnomatematièkog fakulteta u Zagrebu, odr¾ao je niz predavanja u Zagrebu, Splitu i Mostaru. Suradnik je i mentor mnogim hrvatskim i stranim matematièarima. Dopisni je èlan Hrvatske akademije znanosti i umjetnosti. Tijekom rada u Heidelbergu zapa¾ena je njegova pomoæ Hrvatskoj u razlièitim oblicima.  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 J_{4} : And the inside title reveals to full title of Ivanov's monograph:
 Professor Zvonimir Janko at the University of Zagreb, 2007
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 20002003 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 pgourps. 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 [Æepuliæ]. 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 Bo¾ikov, 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 [©iftar]. 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 [Æepuliæ]. 
Math 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
 HallJanko group
 HallJanko graph
 HigmanJankoMcKay group
  For a more detailed description of life and work of Zvonimir Janko see www.croatianhistory.net/etf/janko
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