Re: Finite Group
From: Zbigniew Fiedorowicz (fiedorow_at_hotmail.com)
Date: 03/18/05
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Date: Fri, 18 Mar 2005 07:40:38 -0500
William Elliot wrote:
> Let G be a group for which every proper subgroup is finite.
> Is G finite?
>
Let p be a prime. Consider the multiplicative group of p-th
power roots of unity.
> One observes all elements of G have finite order.
> Pick g /= e and let H = a maximal subgroup not containing g.
> H is finite. <g,H> the group generated by g and G, equals G.
>
No.
> When G is Abelian, then G = <g,H> = { ng + h | n in Z, h in H }
> Now as order of g is finite as also H, <g,H> is finite.
> In fact |G| = |<g,H>| = o(g) * |H|. So far, so good.
>
> What however, if G isn't Abelian?
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