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Zvonimir Janko and Yakov Berkovich authors of important math monograph
By Prof.Dr. Darko Zubrinic | Published  09/21/2009 | Science | Unrated
Croatian and Israeli mathematician working together

Zvonimir Janko, distinguished Croatian mathematician, professer emeritus of the University of Heidelberg, Germany, addressing to the audience at the University of Zagreb during the conference organized in his honour on the occasion of his 75th birthday in 2007.

An important monograph issued in 2008, on 520 pp.


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 Croatian mathematician 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many substantially new results that Berkovich postponed issuing the book, since otherwise the book would be immedately outdated.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 is planned to be issued in three parts, starting with 2008. The second and third part will be a joint work of professors Berkovich and Janko. The second part has been published in 2008 in a voluminous hardcover monograph on 520 pp.

Although at the age nearly 80 years, professor Janko is still amazingly active as a scientist.

Saunders MacLane, distinguished American mathematician, placed Zvonimir Janko among originators of the theory of p-groups.


More information about the monograph

Croatian readers should be aware of a relatively recent monograph devoted to Janko's Fourth Group J4:

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:


Sensational discovery of  The First Janko Group in 1964

To illustrate the "earthquake" in group theory which raised Janko's 1964 discovery of J1 we cite the following testimony from [Ćepulić] (translated from Croatian by D.®.). 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.®.)

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



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


Formated for CROWN by prof.dr. Darko ®ubrinić
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