Math Is Fun Forum

noemi

Member

Registered: 2010-01-07

Posts: 2,333

simple groups

Such a group

has 1 or 4 Sylow 3-subgroups, each of order 9. Suppose 4 and let

be one of them. Then

and so by Humphreys’s corollary, there is a normal subgroup

of

contained in

such that 4 divides

and

divides 4! = 24.

showing that

is not simple.

Wow, it really is simple when you know how.

why is|G:N_G(P)|=4?
what does Humpreys's corollary say? also if u can give me a proof

thank you

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: simple groups

Are you familiar with the theory of group actions? All Sylow p-subgroups of a group

are conjugate, so if

is the set of all Sylow p-subgroups, then

acts on

by conjugation and for any

the number of Sylow p-subgroups is the size of the orbit of

under this action. By the orbit–stabilizer theorem, the size of the orbit of

is the index in

of the stabilizer of

in

, and the stabilizer of

in

is easily seen to be the normalizer of

in

.

For Humphreys’s corollary, see this: http://www.mathisfunforum.com/viewtopic … 38#p109938.

noemi

Member

Registered: 2010-01-07

Posts: 2,333

Re: simple groups

very nice, tnx Jane!