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Math Help - challenging group theory problem

  1. #1
    Apr 2011

    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=eand 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=gfor 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 ( 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.
    Last edited by paragrafo; April 4th 2011 at 06:38 PM.
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