8. (i) Define a permutation of a set X. Let fand g be the permutations of X =

{0, 1, 2, 3, 4, 5, 6} given by f(x) = x+3 and g(x) = 3x, where each is evaluated

modulo 7. Write f and g as products of disjoint cycles. Also express f², g², fg.

and gf as products of disjoint cycles, and for each of f, g, f², g², fg, and gf.

determine whether it is even or odd.

(ii) Prove that the mapping from a group G to itself given by 0(g)

homomorphism if and only if G is abelian.

= g² is a

(iii) State the first isomorphism theorem for groups.

By calculating the kernel and image of the map given in part (ii), deduce that

R*/{1, -1} = R+, where R* and R+ are the sets of non-zero and positive reals

respectively (and the operation is multiplication).