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math8
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Let G=Z_2XS_3 (Z_2:cyclic group of order 2; S_3: Symmetric group on 3) . Show Center of G, Z(G) is not a fully characteristic (or invariant) subgroup of G.
Apparently, Z(G)=Z_2
I know that I need to show that there exists an endomorphism g from G to G such that g(Z_2) is not contained in Z_2.
But I am not sure how.
Also, to prove that every fully characteristic subgroup H is also characteristic, I now how to show that for every automorphism p in Aut(G), p(H) is contained in H, but for some reason I don't see why H is contained in p(H).
Apparently, Z(G)=Z_2
I know that I need to show that there exists an endomorphism g from G to G such that g(Z_2) is not contained in Z_2.
But I am not sure how.
Also, to prove that every fully characteristic subgroup H is also characteristic, I now how to show that for every automorphism p in Aut(G), p(H) is contained in H, but for some reason I don't see why H is contained in p(H).