-group

Given a Prime number , a -group is a group in which the order of every element is an power of , #m/def/group i.e. for all

for some . By Cauchy's order theorem, for a finite group this is equivalent to the order of being an -power of , i.e.

for some .¹

Properties

A nontrivial normal subgroup of a finite -group always has a nontrivial intersection with the centre.²

Proof of 1.

Consider the action of on by conjugation. The orbits of size 1 are the elements of . By the Orbit-stabilizer theorem, the size of orbits divide , hence all orbits have size for some . On the other hand, the ground, divides since the order of a subgroup divides the order of a group. Since is nontrivial, for some . Now adding the sizes of orbits,

so there must be at least one non-identity orbit of size 1, i.e. at least one other central element.