(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Prove the number of distinct conjugate subgroups of a subgroup H in a group G is [G:N(H)] where N(H)={g [itex]\in[/itex] G | gHg[itex]^{-1}[/itex]=H}.

2. Relevant equations

I'm thinking the counting formula; G=|C(x)||Z(x)| with C(x) being the conjugacy class of x and Z(x) being the centralizer of x.

3. The attempt at a solution

I thought that I could say N(H) is the centralizer of H and [G:N(H)]=|G|/|N(H)| so by the counting formula |G|/|N(H)|=|C(H)| and |C(H)| is the number of distinct conjugate subgroups H. In truth I have only worked with the counting formula for elements, never with subgroups. Also if this does hold somewhat true, it seems to me to only apply to the case when G is finite. What about when G is infinite?

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# Normalizer of a subgroup

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