Regular p-group

A finite p-group G is said to be regular if any of the following equivalent (Hall 1959, Ch.

Every p-group of odd order with cyclic derived subgroup is regular.

The subgroup of a p-group G generated by the elements of order dividing pk is denoted Ωk(G) and regular groups are well-behaved in that Ωk(G) is precisely the set of elements of order dividing pk.

In a regular group, the index [G:℧k(G)] is equal to the order of Ωk(G).

In fact, commutators and powers interact in particularly simple ways (Huppert 1967, Kap III §10, Satz 10.8).