Let G be a group of order 3825. Prove that if H is a normal subgroup of order 17 in G, then H is a subgroup of Z(G).

Z(G) = { g in G | xg = gx for every x in G}

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- Nov 3rd 2008, 06:07 PMdori1123abstract algebra
Let G be a group of order 3825. Prove that if H is a normal subgroup of order 17 in G, then H is a subgroup of Z(G).

Z(G) = { g in G | xg = gx for every x in G} - Nov 4th 2008, 04:48 AMNonCommAlg
__Lemma__: both equations and have the unique solution

__Proof__: if then hence but thus: a similar argument works for

now suppose and let be the normal subgroup of G with in order to show that , we only need to prove that .

let we want to show that commutes with let since H is normal, there exists an integer such that thus: which gives us

we have for some since we may assume that now by the__Lemma__we must have

and thus therefore i.e.

__Exercise__: (very easy!) make the problem interesting by extending this result to a class of finite groups! - Nov 4th 2008, 12:55 PMThePerfectHacker
Here is another way (not that there is anything bad about what you did) for

**dori**.

If you have the equation and then where are primes. This is because is a group. And if and then it means has order . By Lagrange's theorem it means divides a contradiction.

Quote:

thus therefore i.e.