Monomorphism categories associated with symmetric groups and parity in finite groups

Monomorphism categories of the symmetric and alternating groups are studied via Cayley's Embedding Theorem. It is shown that the parity is well defined in such categories. As an application, the parity in a finite group G is classified. It is proved that any element in a group of odd order is always even and such a group can be embedded into some alternating group instead of some symmetric group in the Cayley's theorem. It is also proved that the parity in an abelian group of even order is always balanced and the parity in an nonabelian group is independent of its order.