Question: if f and g in Sn are disjoint cycles fg = gf

Answer:

f = (a1 a2 a3... an)

g = (b1 b2 b3 ... bm)

where ai != bj for any i or j

gf(ai) where i = 1 to n-1

gf(ai) = g(ai+1) = ai+1

fg(ai) = f(ai) = ai+1

thus f and g commute for ai where i = 1 to n-1 this same method can be used to show that f and g commute for an and for bi.

Is there a better way to prove this or to think about it?

Thanks!