Explain it briefly

1. Let G be non-abelian group. Show that for all x E G, Z(G) o(Z(G)) + 1. 9. Let N be normal in G. Show that either cl(a) n N = q or cl(a) c N for all a EG. 10. Let N 0. Let N be normal in G, (N # {e}). Show that N ~ Z(G) # {e} 12. If G is finite non-abelian group such that G is abelian, then show that Z (G) k (G) 2 0 G (Hint : Use Problem 28) Z (G) + 0( Z (G) ) - 1 . 13. Show that permutations in S, commuting with o = (1 2 ..., n) are 6, 0' ,..., "-1, 0" =1. 14. Show that permutations in S, commuting with o = (1 2 ..., r), 1