Title: On finite simple groups
Date & Time: Tuesday, March 25 2025, 5:30 PM (Snacks will be available from 5:00 PM - 5:30 PM)
Venue : EEG 301 (Third Floor, GG Building)
Speaker: Prof. NS Narasimha Sastry (Retd professor of Mathematics from Indian Statistical Institute Bangalore Centre)
Abstract:
Finite simple groups are the basic objects all finite groups are composed of. This means that, like prime decomposition of integers, any finite group is made up of a precise finite set (with multiplicity) of finite simple groups. However, unlike prime decomposition of integers, a given finite set of finite simple groups (with specified multiplicities) does not usually determine the group they are the composition factors of. Our ignorance on the problem of determining all finite groups with a given finite set of simple groups (with specified multiplicities) as composition factors is total, except for some small and/or some very specific cases.
The theorem stating that any given finite simple group is isomorphic to one among the known list of finite simple groups is (generally believed to have been) proved by 2004. This humongous and very significant theorem is a consequence of efforts spreading over more than a century, involving major contributions from dozens of mathematicians over decades and the written accounts of various parts spreading over more than 10,000 jornal pages.
The information content of a very simple ( but a very specific!) hypothesis on a finite group, unveiled by relentless logic, includes many marvelous structures ( like, for example, the 26 sporadic simple groups including the most enigmatic monster group) and hitherto unexpected interrelations between seemingly unrelated, significant and exceptional, mathematical entities.
The importance of this work lies in the fact that many- but not all- problems of finite groups can be reduced to questions about finite simple groups and the classification entails a lot of detailed information about each of the simple groups.
A feature of an endeavour such as this is the questions it raises: what does a `proof' of a theorem mean and what is its credibility when no human being can possibly verify the correctness of all the arguments. Given the current feasibility of large groups of scientists collaborating on a particular problem,a la LHC in Physics, it is possible to envisage that in the days to come, many `big' mathematical problems will be attempted (some are work under progress!) which involve the participation of many people, each familiar with only a small fragment of the problem.This raises questions like what a `proof' means in such a situation; how the validity of such a `proof ' be interpreted; what attributes should one look for in a proof (including elegance and clarity, may be); and what should be accepted as truth in mathematics .
In this talk, I try to (very) briefly describe the list of finite simple groups; the mathematical structures they are the symmetries of; some easily statable, but crucial, initial insights that started the classification; some consequences of the classification; and some indication of the marvels of the monster simple group and its connection to modular forms and the affine E_8 root system. Given the enormity of the topic, I will concentrate on easily accessible, but significant, aspects of the topic.