Regular p-group

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, harv|Hall|1932.

Definition

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 "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" · "c"₁^"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" · "c"₁^"q" ⋯ "c"_k^"q", where "q" = "p"^"n".

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.
* 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 "p"² points is irregular and of order "p"^"p"+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.

For every positive integer "k", the elements of order dividing "p"^"k" form a subgroup. In general, 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 exactly the elements of order dividing "p"^"k". The name regular was given due to an even nicer property of the Ω subgroups. The subgroup generated by "p"^"k"th powers of elements in "G" is denoted ℧_"k"("G"). In 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 any normal subgroups "M", "N" of a regular "p"-group "G" and nonnegative integers "m", "n", the following relation holds between powers and commutators: [℧_"m"("M"),℧_"n"("N")] = ℧_"m"+"n"( ["M","N"] ).

Generalizations

* Powerful p-group
* power closed "p"-group

References

*Citation | last1=Hall | first1=Marshall | author1-link=Marshall Hall (mathematician) | title=The theory of groups | publisher=Macmillan | id=MathSciNet | id = 0103215 | year=1959
*Citation | last1=Hall | first1=Philip | author1-link=Philip Hall | title=A contribution to the theory of groups of prime-power order | language=English | year=1933 | journal=Proceedings of the London Mathematical Society, second series | volume=36 | pages=29–95
*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 | id=MathSciNet | id = 0224703 | year=1967 | pages=90–93