Let G be a group, and let h, k be in G, we say that h and k are conjugate if ? g

ID: 2979251 • Letter: L

Question

Let G be a group, and let h, k be in G, we say that h and k are conjugate if ? g in G, such that h = gkg-1. Similarly if H, K are subgroups of G, and ? g in G, such that H = gKg-1, then way we say that H and K are conjugate subgroups. a) Prove that the relation of conjugacy (of elements) is an equivalence relation on G. b) Prove: G is abelian if and only if no element of g is conjugate to any other subgroup of G. c) Prove: if H is a subgroup of G, the gHg-1 is isomorphic to H for all g in G. d) Prove: H is a normal subgroup of G if it is not conjugate to any other subgroup of G. e) Show that all the 2 element subgroups of S3 are conjugate f) Are all the 2 element subgroups of Z

Explanation / Answer

Curious, as this is usually taken to be the definition of normality. I therefore wonder what you are taking to be the definition.

- If it is that all conjugates of N are CONTAINED in N, then, for any g in G, we have

N = 1N1^{-1} = (gg^{-1})N(gg^{-1})^{-1} = g(g^{-1}Ng)g^{-1} contained in gNg^{-1}

- If it is that N is the kernel of some homomorphism on G, then it is easy to see that the conjugates of N are contained in N, and we can thus use the previous observation.

In any case, for the latter claim, it is enough to show that conjugation preserves order (i.e., that conjugation is an isomorphism).

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Let G be a group, and let h, k be in G, we say that h and k are conjugate if ? g

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