Regular p-group
A -group
such that for all
and any integer
an equality
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holds, where are elements of the commutator subgroup of the subgroup generated by the elements
and
. Subgroups and quotient groups of a regular
-group are regular. A finite
-group
is regular if and only if for all
,
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where is an element of the commutator subgroup of the subgroup generated by
and
.
The elements of the form ,
, in a regular
-group
form a characteristic subgroup,
, and the elements of order at most
form a fully-characteristic subgroup,
.
Examples of regular -groups are
-groups of nilpotency class at most
, and
-groups of order at most
. For any
, there is a non-regular
-group
of order
(it is isomorphic to the wreath product of the cyclic group of order
with itself).
References
[1] | M. Hall, "Group theory" , Macmillan (1959) |
Regular p-group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Regular_p-group&oldid=14502