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 .

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 ap ÃÂ· bp = (ab)p ÃÂ· 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 ÃÂ· bp = (ab)p ÃÂ· c1p Ã¢ÂÂ¯ ckp.

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 ÃÂ· bq = (ab)q ÃÂ· c1q Ã¢ÂÂ¯ ckq, 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 HallÃ¢ÂÂPetresco identity.

Every p-group of order at most pp 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 pp+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 pk is denoted ÃÂ©k(G) and regular groups are well-behaved in that ÃÂ©k(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 Ã¢ÂÂ§k(G). In a regular group, the index [G:Ã¢ÂÂ§k(G)] is equal to the order of ÃÂ©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 [Ã¢ÂÂ§m(M),Ã¢ÂÂ§n(N)] = Ã¢ÂÂ§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:Ã¢ÂÂ§1(G)] < pp

# [':Ã¢ÂÂ§1(')| < ppÃ¢ÂÂ1

Generalizations

Powerful p-group

power closed p-group

References