Can we not just apply the Burnside Basis Theorem and prove that all groups of order

are Abelian? This would clearly prove the result.

Let

be a finite

-group, and denote by

the derived subgroup,

.

Denote by

the Frattini subgroup; the intersection of all the maximal subgroups of

. Then

is an elementary Abelian group which we can view as a vector space of rank

. The Burnside Basis Theorem states:

i)

.

ii) For

such that

then there exists

such that

and

.

iii) Another statement that isn't really relevant here but basically says that the generators of the group are precisely the coset representatives for the generators of the quotient group.

Thus, if

we have that the group is cyclic (by part (ii) ), and if

then we have that

and the group is Abelian (by part (i) ). Clearly there exists a subgroup of order

in the group so the group cannot have

. Thus the group is Abelian.

I know it is a bit like overkill, but it is my favorite theorem...