Regular p-group

A -group such that for all and any integer an equality

holds, where are elements of the commutator subgroup of the subgroup generated by the elements and . Subgroups and quotient groups of a regular -group are regular. A finite -group is regular if and only if for all ,

where is an element of the commutator subgroup of the subgroup generated by and .

The elements of the form , , in a regular -group form a characteristic subgroup, , and the elements of order at most form a fully-characteristic subgroup, .

Examples of regular -groups are -groups of nilpotency class at most , and -groups of order at most . For any , there is a non-regular -group of order (it is isomorphic to the wreath product of the cyclic group of order with itself).

References

[1]	M. Hall, "Group theory" , Macmillan (1959)