>>11963825Let be a nontrivial element, if its order is we're done
so its order is , and there is a nontrivial element in but not generated by
If the order of is then the group is cyclic, so its also
Since has order , it is normal in , since its index is , which is the smallest prime dividing
so the subgroup product is also a subgroup
and now
since intersect is a subgroup its order divides , and since is not in , its order is strictly less than
so its just trivial
and
so every element of can be written uniquely as
likewise we could write everything as a power of then a power of
since is normal, , and since is normal
then so
by uniqueness and , we have
so
so now theres clearly an isomorphism to
god, arent finite groups so elegant
what lovely counting arguments