I'd like to know how to prove these permutation cycles identities:
$(x, y)(x, y) = id \text{, where id is the identity permutation}\\ (x, y)(x, z) = (x, z, y)\\ (i_1,i_2,i_3,...,i_{k-1},i_k)=(i_1,i_k)(i_1,i_{k-1})...(i_1,i_3)(i_1,i_2)\\ \alpha(i_1,i_2,...,i_k)\alpha^{-1}=(\alpha(i_1),\alpha(i_2),...,\alpha(i_k))\text{, where }\alpha\text{ is an arbitrary permutation (I think so)} $
I've proved the first three by multiplying the cycles (is there a better way?), but I don't know how to show the last one.