can someone explain what is meant by a Sylow p-subgroup

and hence outline the steps in the following question:

"Let G be a group. Prove that if G has only one Sylow p-subgroup H, then H is normal in G."

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- June 17th 2012, 03:38 PMibnashrafSylow p-subgroup
can someone explain what is meant by a Sylow p-subgroup

and hence outline the steps in the following question:

"Let G be a group. Prove that if G has only one Sylow p-subgroup H, then H is normal in G." - June 17th 2012, 04:12 PMibnashrafRe: Sylow p-subgroup
I think one of my problems here is that G was defined to be a group.

If G was defined to be a "finite group", then I could have said that by definition H is a subgroup of G.

However, there is no finite group mentioned in the question, so how exactly am I to proceed ? - June 17th 2012, 06:31 PMHallsofIvyRe: Sylow p-subgroup
- June 17th 2012, 06:41 PMibnashrafRe: Sylow p-subgroup
- June 18th 2012, 06:49 AMDevenoRe: Sylow p-subgroup
yes, you have to show that for any g in G, and h in H, that ghg

^{-1}is in H.

equivalently (and easier to use for this problem), you need to show that gHg^{-1}is a subset of H, for any g in G.

some things to think about:

a) prove that for any subgroup H of G, and any element g in G, the set gHg^{-1}= {ghg^{-1}: h in H} is actually a subgroup of G.

b) show that H and gHg^{-1}have the same order.

conclude that if G has only ONE subgroup of a given order, that subgroup MUST be normal (sylow subgroup or not).

the definition of a sylow p-subgroup is, by the way:

a subgroup of order p^{k}, where p^{k}divides |G|, and p^{k+1}does not divide |G|.

for example, if G is a group of order 24, a sylow 2-subgroup of G is a subgroup of order 8, whereas a sylow 3-subgroup is a subgroup of order 3.