regular p-group

A finite p-group G is said to be regular if any of the following equivalent {{harv|Hall|1959|loc=Ch. 12.4}}, {{harv|Huppert|1967|loc=Kap. III §10}} conditions are satisfied:

• For every a, b in G, there is a c in the derived subgroup {{prime|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.

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.

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 Ω_k(G) 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). 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 {{harv|Huppert|1967|loc=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]).

• Philip Hall's criteria of regularity of a p-group G: G is regular, if one of the following hold:
• # [G:℧₁(G)] < p^p
• # [{{prime|G}}:℧₁({{prime|G}})| < p^p−1
• # |Ω₁(G)| < p^p−1

• Powerful p-group
• power closed p-group

• {{Citation | last1=Hall | first1=Marshall | author1-link=Marshall Hall (mathematician) | title=The theory of groups | publisher=Macmillan |mr=0103215 | year=1959}}
• {{Citation | last1=Hall | first1=Philip | author1-link=Philip Hall | title=A contribution to the theory of groups of prime-power order | year=1934 | journal=Proceedings of the London Mathematical Society | volume=36 | pages=29–95 | doi=10.1112/plms/s2-36.1.29}}
• {{Citation | last1=Huppert | first1=B. | author1-link=Bertram Huppert | title=Endliche Gruppen | publisher=Springer-Verlag | location=Berlin, New York | language=German | isbn=978-3-540-03825-2 | oclc=527050 |mr=0224703 | year=1967 | pages=90–93}}

Category:Properties of groups

Category:Finite groups

Category:P-groups