(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 D_n(G) is normal in all of G not just in D_n-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 S_n are inner, except for S₆ which does have an outer automorphism.