Let G be a group (not necessarily finite). A subgroup S ≤ G is said to be characteristic if σ(S) ≤ S for every σ ∈ Aut(G). (a) Prove that every characteristic subgroup of G is normal in G (b) Prove that the center Z(G) of G is characteristic. (c) Prove that if S is characteristic in G then σ(S) = S for every σ ∈ Aut(G). (d) Let p be a prime and let P be the subgroup of G generated by all elements of G whose order is a power of p. Prove that P is a characteristic subgroup of G.
Added by Adam C.
Step 1
That is, for any g in G, we need to show that gSg^-1 is a subset of S and S is a subset of gSg^-1. Let σ be an arbitrary automorphism of G. Since S is characteristic, we have σ(S) ≤ S. Now consider the conjugate subgroup gSg^-1. We want to show that gSg^-1 is a Show more…
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