challenging group theory problem

• Apr 4th 2011, 06:13 PM
paragrafo
challenging group theory problem
this problem is taken from the herstein book topics in algebra:
G is a finite group such that n divides o(G) define the set H={x/ x^n=e} (is not always a group), prove that the number of elements in H is a multiple of n.
o(G) denotes the number of elements in G.
so this is what i've done so far:
the problem before this is the special case in which G is abelian and is much easier, if G is abelian H is subgroup of G, then if $p^k$ divides n then it divide s o(G) and by Sylow theorem there exist a subgroup S of order $p^k$ in particular
$a\in S$ implies $a^p^k=e$and $a\in H$, so $S\subset H$ and by lagrange theorem $p^k$ divides n, do this for each p prime divisor of n and we get n divides o(H).
as for the general case all i could do was the following: if p divides n and p divides o(G) then p divides o(H).
the key is to observe that H is closed under conjugacy, this is to say if $h\in H$ then $g^-hg\in H$,so take some fixed element a of order p (existence asserted by cauchy theorem) and break up H by conjugacy classes relative to (a) (the generated by a), this is to say $h\sim g$ iff $b^-hb=g$for some $b\in (a)$,now the same argument as in the class equation $o([h])=\frac{o(a)}{N(h)\cap(a)}$ where
N(h)={x/xh=hx} this is because $b^-hb=c^-hc$ iff $N(h)\cap(a)b=N(h)\cap(a)c$. Now take the sum over h in distinct equivalence classes and take mod p, all nontrivial classes vanishes so that $o(H)=o(N(a)\cap H) mod p$ , if N(a) is not G then by induction over the size of G we have what we seek.
if N(a) is G then more work is needed ((Headbang) i know), H is closed under right translation relative to (a) this is to say $h\in H$ implies $hb\in H$ for $b\in (a)$, this is because every element conmutes with a so that $(ha)^n=h^na^n$, now break up H by left translation relative to (a) this is $g\sim h$ iff $ga^k=h$ this gives as in lagrange theorem p distinct equivalence classes with the same number of elements so that p divides o(H).
I hope someone reads this and manage to give a complete solution, any help is welcome.