# Help proving that a simple group of order 360 must be isomorphic to A_6

*bill-deja_at_weacca.com*

**Date:** 03/16/05

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Date: 16 Mar 2005 04:54:29 -0800

I am looking for a reasonably elementary proof that a simple group of

order 360 must have order 360. I have proven all sorts of things

about the Sylow subgroups, their normalizers, and their intersections;

but I haven't stumbled across anything that gives me the result.

I suspect that it might be fruitful to take a Sylow 5 subgroup P. Its

normalizer contains 5 elements of order 2. Take one of them. It

also belongs to a subgroup H of G isomorphic to A_4. If we could show

that PH is not all of G, we would be done.

Another approach might be to somehow use the fact that if H acts on G's

Sylow 5 subgroups by conjugation then Orbit(P) is of order 6, most

plausibly by somehow using this fact to partition the Sylow 5 subgroups

into 6 sets of 6, and then having G somehow act on these. Each

approach is believable because it's easy to make it work in A_6 itself.

But I cannot see how to get either of them to work.

Any hints would be appreciated --- either on how to make one of these

ideras work out --- or on some other approach.

BTW, the subgroups of order 12 are interesting. Some of them clearly

correspond to two pint stabilizers, e.g. the one generated by (123) and

(12)(34) clearly stabilizes 4 and 5; but I don't know what to make of

the other ones, e.g. the one generated by

(12)(34) and (135)(246).

Bill.

**Next message:**bill-deja_at_weacca.com: "Re: How to attack this without Sylow's thm"**Previous message:**Ibn.Qirtaiba_at_gmail.com: "graphs of the regular polytopes"**Next in thread:**David C. Ullrich: "Re: Help proving that a simple group of order 360 must be isomorphic to A_6"**Reply:**David C. Ullrich: "Re: Help proving that a simple group of order 360 must be isomorphic to A_6"**Reply:**mareg_at_mimosa.csv.warwick.ac.uk: "Re: Help proving that a simple group of order 360 must be isomorphic to A_6"**Reply:**Jim Heckman: "Re: Help proving that a simple group of order 360 must be isomorphic to A_6"**Messages sorted by:**[ date ] [ thread ]