Let f,g be distinct transpositions.

a) Show that if f and g are disjoint, then fg can be expressed as the product of two 3-cycles.

Proof:

Let f=(a,b),g=(c,d)→fg=(a,b)(c,d)=(a,b,c,d)=(a,b,c)(a, b,d)

Therefore, fg can be expressed as the product of two 3-cycles.

b) Show that if f and g are not disjoint, the fg can be expressed as a 3-cycle.

Proof:

Let f=(a,b),g=(b,d)→fg=(a,b,d)

Therefore, fg can be expressed as a 3-cycle.