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), (Huppert 1967) 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 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 · c₁^p ⋯ c_k^p.
• 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 · c₁^q ⋯ c_k^q, where q = pⁿ.

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 Ω_k(G) and regular groups are well-behaved in that Ω_k(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 Ω_k(G). In fact, commutators and powers interact in particularly simple ways (Huppert 1967). 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)] < p^p
  2. [G′:℧₁(G′)| < p^p−1
  3. |Ω₁(G)| < p^p−1

Generalizations

• Powerful p-group
• power closed p-group

References

• Hall, Marshall (1959), The theory of groups, Macmillan 
• Hall, Philip (1934), "A contribution to the theory of groups of prime-power order", Proceedings of the London Mathematical Society 36: 29–95, doi:10.1112/plms/s2-36.1.29 
• Huppert, B. (1967) (in German), Endliche Gruppen, Berlin, New York: Springer-Verlag, pp. 90–93, ISBN 978-3-540-03825-2, OCLC 527050