Let v be a permutation of n elements such that $v^2$ is the identity.
Prove that $v$ is a product of disjoint transpositions.
My thinking: $v^2$ is the identity so intuivitly I can see why it is true. Since after $V$ is applied, you have then have to do the exact opposite to get by the identity element. (by applying $V$), which implies a product of disjoint transpositions.
But how would I go about an actual proof?
The permutation $v$ can have only cycles of length $1$ or $2$, otherwise the order of the permutation is larger than $2$, so $v^2$ cannot be the identical permutation.