《Taming the Monster(Wilson)》.pdf

The taming of the Monster Robert A Wilson Wednesday 4th December 2002 Auckland Groups are the abstract embodiment of sym- metry in mathematics, and as such their study is of fundamental importance wherever sym- metry can be used to simplify problems. The trouble is that groups are complicated objects, despite their simple definition, and a complete classification of them is hopeless. We can however sometimes break them down into smaller groups: if N is a normal subgroup of a group G, then the set G/N = {Ng : g ∈ G} of cosets Ng = {ng : n ∈ N } forms a quotient group. In these circumstances, we can study G by studying the smaller groups N and G/N , and the way they are stuck together to form G. If we cannot break the group down in this way, it is called simple. Clearly a group of prime order is simple, as it has no subgroups at all. All other simple groups are non-abelian (i.e. non-commutative). Galois initiated the study of non-abelian simple groups in the 1830s: the fact that the alternat- ing group A5 is simple proves that the general quintic equation is not soluble by radicals. He found a number of other simple groups, including what we now call PSL2 (p ), for primes p ≥ 5. The obvious question is, can we find all the finite simple groups? Well, to cut a long story short . . . CFSG the Classification theorem for Finite Simple Groups Every finite simple group is either: • a