Consider the following mappings between a permutation group and a cyclic group.
(a) Denote by {A}_{n} the subset of the permutation group {S}_{n} that contains all the even permutations. Show that {A}_{n} is a subgroup of {S}_{n}.
(b) List the elements of {S}_{3} in cycle notation and identify the subgroup {A}_{3}.
(c) For each element X of {S}_{3}, let p(X) = 1 if X belongs to {A}_{3} and p(X) = −1 if it does not. Denote by {C}_{2} the multiplicative cyclic group of order 2. Determine the images of each of the elements of {S}_{3} for the following four mappings:
{Φ}_{1} : {S}_{3} → {C}_{2} X → p(X),
{Φ}_{2} : {S}_{3} → {C}_{2} X →−p(X),
{Φ}_{3} : {S}_{3} → {A}_{3} X → {X}^{2},
{Φ}_{4} : {S}_{3} → {S}_{3} X → {X}^{3}.
(d) For each mapping, determine whether the kernel K is a subgroup of {S}_{3} and if so, whether the mapping is a homomorphism.