High Quality Content by WIKIPEDIA articles! In mathematics, especially in the field of 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. The concept was first described by Philip Hall in his foundational contribution to p-groups, (Hall 1932). The subgroup of a p-group G generated by the elements of order dividing pk is denoted ?k(G) and regular groups are well-behaved in that ?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 [G:?k(G)] is equal to the order of ?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(M),?n(N)] = ?m+n([M,N]).

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