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

I'm trying to prove that, if H is a subgroup of an arbitrary group G, then H^g, the action of a given element in G on H, is isomorphic to H.

2. Relevant equations

3. The attempt at a solution

Let \sigma denote a given action of G on H. We are considering the map \sigma(g, *) : H -> H^g (where g is fixed and * denotes a variable element of H). I think that the kernel of this map is the set of all elements x in H such that \sigma(g, x) = g (this is the part I'm not sure of). Since x lies in H and H is a subgroup of G, it follows that x must be equal to e, the identity element of G. Thus, ker{\sigma(g, x)} is trivial, and, as a result,

[tex] H^g \cong G/ \mbox{ker}(\sigma(g, *)) = G, [/tex]

by the first isomorphism theorem.

Does this seem correct? If so, can you please justify the statement that

[tex] \mbox{ker} \sigma(g, *) = \{ x| \sigma(g, x) = g \ \forall x \in H \} [/tex]?

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# Homework Help: Group action on a subgroup

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