Regular p-group

A $ p $- group $ G $ such that for all $ a , b \in G $ and any integer $ n = p ^ \alpha $ an equality

$$ ( a b ) ^ {n} = a ^ {n} b ^ {n} s _ {1} ^ {n} \dots s _ {t} ^ {n} $$

holds, where $ s _ {1} \dots s _ {t} $ are elements of the commutator subgroup of the subgroup generated by the elements $ a $ and $ b $. Subgroups and quotient groups of a regular $ p $- group are regular. A finite $ p $- group $ G $ is regular if and only if for all $ a , b \in G $,

$$ a ^ {p} b ^ {p} = ( a b ) ^ {p} s ^ {p} , $$

where $ s $ is an element of the commutator subgroup of the subgroup generated by $ a $ and $ b $.

The elements of the form $ a ^ {p ^ \alpha } $, $ a \in G $, in a regular $ p $- group $ G $ form a characteristic subgroup, $ C ^ \alpha ( G) $, and the elements of order at most $ p ^ \alpha $ form a fully-characteristic subgroup, $ C _ \alpha ( G) $.

Examples of regular $ p $- groups are $ p $- groups of nilpotency class at most $ p - 1 $, and $ p $- groups of order at most $ p ^ {p} $. For any $ p $, there is a non-regular $ p $- group $ S ( p ^ {2} ) $ of order $ p ^ {2} $( it is isomorphic to the wreath product of the cyclic group of order $ p $ with itself).

References

[1]	M. Hall, "Group theory" , Macmillan (1959)