On Tue, 23 Sep 2003, Richard Schroeppel wrote:
John, could you state precisely the unique factorization theorem? I've never seen it written down. There seem to be some subtleties, such as checking that AxB = AxC implies B=C. (For groups, of course, we interpret "=" as isomorphism.) Is it also true for infinite groups?
The easiest (because weakest) statement is that every finite group is a direct product of indecomposable groups, and that the isomorphism types of these groups are uniquely determined up to order. ["Indecomposable" means "not a direct product of two non-trivial groups"]. There are stronger forms, but I can never remember them, and find it dangerous to try, because so many possible strengthenings are wrong. For example the groups C2 and D8 (the latter meaning the dihedral group of order 8) are indecomposable, but there are four (or maybe 8) different pairs of subgroups of types C2 and D8 in C2 x D8 of whioch it's the direct product. Regards, John Conway