In mathematical finite group theory, the concept of regular p-group captures some of the more important properties of abelian

Abelian group

In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order . Abelian groups generalize the arithmetic of addition of integers...

 p-group

P-group

In mathematics, given a prime number p, a p-group is a periodic group in which each element has a power of p as its order: each element is of prime power order. That is, for each element g of the group, there exists a nonnegative integer n such that g to the power pn is equal to the identity element...

s, 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 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
  Abelian group
  In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order . Abelian groups generalize the arithmetic of addition of integers...
  
   p-group is regular.
• Every p-group of nilpotency class strictly less than p is regular.
• Every p-group of order
  Order (group theory)
  In group theory, a branch of mathematics, the term order is used in two closely related senses:* The order of a group is its cardinality, i.e., the number of its elements....
  
   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
  Symmetric group
  In mathematics, the symmetric group Sn on a finite set of n symbols is the group whose elements are all the permutations of the n symbols, and whose group operation is the composition of such permutations, which are treated as bijective functions from the set of symbols to itself...
  
   on p² points is irregular and of order p^p+1.

Properties

A p-group is regular if and only if

If and only if

In logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements....

 every subgroup

Subgroup

In group theory, given a group G under a binary operation *, a subset H of G is called a subgroup of G if H also forms a group under the operation *. More precisely, H is a subgroup of G if the restriction of * to H x H is a group operation on H...

 generated by two elements is regular.

Every subgroup and quotient group

Quotient group

In mathematics, specifically group theory, a quotient group is a group obtained by identifying together elements of a larger group using an equivalence relation...

 of a regular group is regular, but the direct product

Direct product of groups

In the mathematical field of group theory, the direct product is an operation that takes two groups and and constructs a new group, usually denoted...

 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

Cyclic group

In group theory, a cyclic group is a group that can be generated by a single element, in the sense that the group has an element g such that, when written multiplicatively, every element of the group is a power of g .-Definition:A group G is called cyclic if there exists an element g...

. 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)

Omega and agemo subgroup

In mathematics, or more specifically group theory, the omega and agemo subgroups described the so-called "power structure" of a finite p-group. They were introduced in where they were used to describe a class of finite p-groups whose structure was sufficiently similar to that of finite abelian...

 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)

Omega and agemo subgroup

In mathematics, or more specifically group theory, the omega and agemo subgroups described the so-called "power structure" of a finite p-group. They were introduced in where they were used to describe a class of finite p-groups whose structure was sufficiently similar to that of finite abelian...

. 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 . 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
  Philip Hall
  Philip Hall FRS , was an English mathematician.His major work was on group theory, notably on finite groups and solvable groups.-Biography:...
  
  '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