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(a) Find an example of groups G, H, K with K H and H G but K G.
(b) A subgroup H of G is characteristic if σ(H) ⊆ H for every group automorphism σ of G. Show that every characteristic subgroup is normal.
(c) Show that if K ⊆ H ⊆ G, K is characteristic in H and H is characteristic in G then K is characteristic in G.
(d) Show that Dn(G) is normal in all of G not just in Dn-1(G).
(e) Conclude that G is solvable i it has a subnormal series in which each quotient is abelian and each group in the series is normal in all of G.
If G is a group, an automorphism of the form is called an inner automorphism. An automorphism not of this form is called outer.
The aim of the following questions is to show that all the automorphisms of Sn are inner, except for S6 which does have an outer automorphism.
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