Comments on: A look at a few Tripos questions X

By: robin harte

robin harte — Thu, 09 Aug 2012 00:48:20 +0000

talking about “normal subgroups” of groups leads me to the idea of “normal” subgroups of semigroups – start with “generalized exponentials” in Banach algebras, they form the connected component of the identity in the invertible group: so what can you say about their left and right cosets in the larger semigroups of left and of right invertibles ? you can still multiply

I am being uncareful here and not checking what I have looked at elsewhere, but the idea is a very simple extension of the usual quotient group idea, and is involved in a discussion of “spectral pictures”

By: Stones Cry Out - If they keep silent… » Things Heard: e223v1n2

Stones Cry Out - If they keep silent… » Things Heard: e223v1n2 — Tue, 29 May 2012 13:22:45 +0000

[...] The test approaches, fun with groups. [...]

By: Monday/Tuesday Highlights | Pseudo-Polymath

Monday/Tuesday Highlights | Pseudo-Polymath — Tue, 29 May 2012 13:22:12 +0000

[...] test approaches, fun with [...]

By: Gareth

Gareth — Tue, 29 May 2012 13:04:32 +0000

The general waffle I often use at the start of the n! part of the Groups question is as follows. Since my supervisees have found it useful, I thought I’d add it here.

Suppose the group G acts on the set X, and that there are more elements in G than there are things we can actually do to X. Then, by the pigeonhole principle, some two things in G must have exactly the same effect on X. So doing one followed by the inverse of the other does nothing to X.

Aha, something does nothing – so there is a non-trivial kernel (to some homomorphism). Now, is the kernel all of G? If not, then we have a non-trivial proper normal subgroup.

So, if G does anything at all, and if G is bigger than |X|!, then G can’t be simple.