Regular p-group

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:
  1. [G:℧1(G)] p
  2. :℧1()| p−1
  3. |Ω1(G)| p−1

Generalizations

• [Powerful p-group
• power closed p-group

References