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