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}
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!
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.
You mean . But whatever it is not any significant.thus therefore i.e.