1. Let G be a finite abelian group of order o(G) and suppose the integer n is relatively prime to o(G). Prove that every g ∈ G can be written as g = xn with x ∈ G. (Hint: Consider the mapping f: G ~ G defined by f(y) = yn, and prove this mapping is an isomorphism of G onto G.) Proof: an = bn gives (ab-1)n = e. If xp = e, then |x| is the divisor of p. Hence |ab-1| is a divisor of n. By Lagrange’s Thm it is also a divisor of |G|. Hence its a common divisor of n and |G|, so |ab-1| = 1 so ab-1 = e. 9. Prove that the center of a group is a normal subgroup. Proof: Z(g)={g ∈ G ga=ag ∀a ∈ G}. It is straightforward to show that Z(G) is a subgroup of G. eg=ge, so e ∈ Z(g); Let a,b ∈ Z(g), we know that ∀g ∈ G: ag = ga, bg = gb. Thus gab = agb = abg, hence ab ∈ Z(g); Let c ∈ Z(g), we know that ∀g ∈ G: cg = gc, thus c-1gcc-1 = c-1cgc-1 => c-1g = gc-1 hence c-1 ∈ Z(g). To show it is normal it suffices to show that if g ∈ Z(g) then aga−1 ∈ Z(G) for all a∈ G . But aga−1 = aa−1g = g, since g commutes with all elements of G. Thus aga−1 ∈ Z(G) for all a∈G. 17. Let G be a group and Z the center of G. If is any automorphism of G, prove that (Z)T ⊂ Z. 4. (a) If H is a subgroup of G show that for every g ∈ G, gHg-1 is a subgroup of G. (b) Prove that W = intersection of all gHg-1 is a normal subgroup of G. 8. If in a finite group G an element a has exactly two conjugates, prove that G has a normal subgroup N =/= (e), G. 10. (a) If N is a normal subgroup of G and a ∈ N, show that every conjugate of a in G is also in N. (b) Prove that o(N) = ∑ca for some choices of a in N. 18. Prove that a group of order 28 has a normal subgroup of order 7. 19. Prove that if a group G of order 28 has a normal subgroup of order 4, then G is abelian.