View source for Regular p-group

<!--
r0807702.png
$#A+1 = 33 n = 0
$#C+1 = 33 : ~/encyclopedia/old_files/data/R080/R.0800770 Regular \BMI p\EMI\AAhgroup
Automatically converted into TeX, above some diagnostics.
Please remove this comment and the {{TEX|auto}} line below,
if TeX found to be correct.
-->

{{TEX|auto}}
{{TEX|done}}

A [[P-group| $  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|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|characteristic subgroup]],  $  C  ^  \alpha  ( G) $, 
and the elements of order at most  $  p  ^  \alpha  $
form a [[Fully-characteristic subgroup|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|wreath product]] of the cyclic group of order  $  p $
with itself).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  M. Hall,   "Group theory" , Macmillan  (1959)</TD></TR></table>