2. (a) Determine, giving reasons, which of the following groups are cyclic.

i. The subgroup pZ of Z where p is a prime and the operation is addition.

ii. The direct product K = Z5 × Z7 with the usual operation on direct products.

iii. The direct product K = Z21 XZ49 with the usual operation on direct products.

iv. The subgroup T = {z € C* : |z| = 1} of (C*, x).

(b) i. Write the following permutations in the symmetric group S10 as products of

disjoint cycles:

f

g=

1 2 3 4 5 6 7 8 9 10

2 3 7 6 8 9 1 10 4 5

1 2 3 4 5 6 7 8 9

3 6 10 1 5 7 4 2 8

9).

10

9

Calculate f-¹ and fg, as products of disjoint cycles.

ii. State a criterion for members of a finite symmetric group to be conjugate, and

determine which of f, g, f¹, fg from part (i) are conjugate.

(c) Let G be a group and : G→ G be a map defined by y(g) = g-¹.

i. Prove that is an bijection.

ii. Prove that

is an isomorphism if and only if G is Abelian.