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 ''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. 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 ''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.

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 :℧_''k''(''G'')is equal to the order of Ω_''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 ��_''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''
*# < ''p''^{''p''−1}
*# , Ω₁(''G''), < ''p''^{''p''−1}

Generalizations

* Powerful p-group
* power closed ''p''-group

References

*
*
*{{Citation "> last1=Huppert , first1=B. , author1-link=Bertram Huppert , title=Endliche Gruppen , publisher= location=Berlin, New York , language=German , isbn=978-3-540-03825-2 , oclc=527050 , mr=0224703 , year=1967 , pages=90–93

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