11. Write out the cycle decomposition of the eight permutations in $4 corresponding to the elements of Dg given by the action of Dg on the vertices of a square (where the vertices of the square are labelled as in Section 2). 12. Assume n is an even positive integer and show that D2, acts on the set consisting of pairs of opposite vertices of a regular n-gon. Find the kernel of this action (label vertices as usual). 13. Find the kernel of the left regular action. 14. Let G be a group and let A = G. Show that if G is non-abelian then the maps defined by g.a = ag for all g, a e G do not satisfy the axioms of a (left) group action of G on itself. 15. Let G be any group and let A = G. Show that the maps defined by g a = ag- for all g, a e G do satisfy the axioms of a (left) group action of G on itself. 16. Let G be any group and let A = G. Show that the maps defined by g a = gag- for all g, a e G do satisfy the axioms of a (left) group action (this action of G on itself is called conjugation). 17. Let G be a group and let G act on itself by left conjugation, so each g E G maps G to G by x-gx8 . For fixed g E G, prove that conjugation by g is an isomorphism from G onto itself (i.e., is an automorphism of G - cf. Exercise 20, Section 6). Deduce that x and gxg- have the same order for all x in G and that for any subset A of G, |A| = 18Ag- | (here gAg = (gag-' [ a e A]). 18. Let H be a group acting on a set A. Prove that the relation ~ on A defined by a~b if and only if a = hb for some he H is an equivalence relation. (For each x e A the equivalence class of x under ~ is called the orbit of x under the action of H. The orbits under the action of H partition the set A.)