Talk:Characteristic subgroup

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I have to argue, the center of the group is not always fully characteristic, there is a group with order 16 as counterexample.

Ooops! Quite right; in fact, let G be the direct product of C₂ and any generalized dihedral group Dih(H), then G forms a counterexample. If we let f:G → G be the endomorphism f(h) = 1 for all h in H, f(g) = g for g not in H, then f(G) is onto the subgroup generated by C₂ and an element of order 2 in Dih(H); this is isomorphic to V₄, so we can apply an automorphism T swapping elements of order 2. Then Tf(Z(G)) is not a subgroup of Z(G).

It is true that the center of a group is strictly characteristic. Chas zzz brown 20:49 Nov 5, 2002 (UTC)

Is strictly characteristic the usual term? I've never seen this term before, but I have seen such subgroups referred to as distinguished subgroups, e.g., in Combinatorial Group Theory (Magnus, Karrass & Solitar). --Zundark 09:07, 7 Oct 2003 (UTC)

It's been over two years since I asked this, and still nobody has provided a reference for "strictly characteristic", so I'm changing it to "distinguished". --Zundark 09:34, 2 November 2005 (UTC)Reply

Section 2 ended with

"Normality" is not transitive but Characteristic is transitive.So when transitivity will hold for Normal Subgroups?? If H Char K and K normal in G then H normal in G.

I removed the fragmentary question because it did not seem to impart any information.–Dan Hoey^talk 16:46, 14 September 2010 (UTC)Reply