Talk:Characteristic subgroup

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 &rarr; 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)

Sentence fragment removed
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 Hoeytalk 16:46, 14 September 2010 (UTC)