# Regular p-group

In mathematical finite group theory, the concept of regular p-group captures some of the more important properties of abelian p-groups, but is general enough to include most "small" p-groups. Regular p-groups were introduced by Phillip Hall (1934).

## Definition

A finite p-group G is said to be regular if any of the following equivalent (Hall 1959, Ch. 12.4), (Huppert 1967, Kap. III §10) conditions are satisfied:

• For every a, b in G, there is a c in the derived subgroup H′ of the subgroup H of G generated by a and b, such that ap · bp = (ab)p · cp.
• For every a, b in G, there are elements ci in the derived subgroup of the subgroup generated by a and b, such that ap · bp = (ab)p · c1pckp.
• For every a, b in G and every positive integer n, there are elements ci in the derived subgroup of the subgroup generated by a and b such that aq · bq = (ab)q · c1qckq, where q = pn.

## Examples

Many familiar p-groups are regular:

However, many familiar p-groups are not regular:

## Properties

A p-group is regular if and only if every subgroup generated by two elements is regular.

Every subgroup and quotient group of a regular group is regular, but the direct product of regular groups need not be regular.

A 2-group is regular if and only if it is abelian. A 3-group with two generators is regular if and only if its derived subgroup is cyclic. 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. The subgroup generated by all pk-th powers of elements in G is denoted k(G). 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). For example, given normal subgroups M and N of a regular p-group G and nonnegative integers m and n, one has [℧m(M),℧n(N)] = ℧m+n([M,N]).

• Philip Hall's criteria of regularity of a p-group G: G is regular, if one of the following hold:
1. [G:℧1(G)] < pp
2. [G′:℧1(G′)| < pp−1
3. 1(G)| < pp−1