Regular P-group - Properties

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 Ω_k(G) and regular groups are well-behaved in that Ω_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 ℧_k(G). In a regular group, the index is equal to the order of Ω_k(G). In fact, commutators and powers interact in particularly simple ways (Huppert 1967, Kap III §10, Satz 10.8). For example, given normal subgroups M and N of a regular p-group G and nonnegative integers m and n, one has = ℧_m+n.

• Philip Hall's criteria of regularity of a p-group G: G is regular, if one of the following hold:
  1. < pp
  2. [G′:℧₁(G′)| < pp−1
  3. |Ω₁(G)| < pp−1