Regular p-group
title: "Regular p-group" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["properties-of-groups", "finite-groups", "p-groups"] topic_path: "general/properties-of-groups" source: "https://en.wikipedia.org/wiki/Regular_p-group" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0
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 .
Definition
A finite p-group G is said to be regular if any of the following equivalent , conditions are satisfied:
- For every a, b in G, there is a c in the derived subgroup ** of the subgroup H of G generated by a and b, such that a**p · b**p = (ab)p · c**p.
- For every a, b in G, there are elements c**i in the derived subgroup of the subgroup generated by a and b, such that a**p · b**p = (ab)p · c1p ⋯ ckp.
- For every a, b in G and every positive integer n, there are elements c**i in the derived subgroup of the subgroup generated by a and b such that a**q · b**q = (ab)q · c1q ⋯ ckq, where q = p**n.
Examples
Many familiar p-groups are regular:
- Every abelian p-group is regular.
- Every p-group of nilpotency class strictly less than p is regular. This follows from the Hall–Petresco identity.
- Every p-group of order at most p**p is regular.
- Every finite group of exponent p is regular.
However, many familiar p-groups are not regular:
- Every nonabelian 2-group is irregular.
- The Sylow p-subgroup of the symmetric group on p2 points is irregular and of order p**p+1.
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 p**k is denoted Ωk(G) and regular groups are well-behaved in that Ωk(G) is precisely the set of elements of order dividing p**k. The subgroup generated by all p**k-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 . 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:
- [G:℧1(G)] p
- [:℧1()| p−1
- |Ω1(G)| p−1
Generalizations
- Powerful p-group
- power closed p-group
References
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