Omega subgroups are variety-containing in regular p-group
Contents
Definition
Suppose is a Regular p-group (?). Then, for any natural number
, the subgroup:
is a Variety-containing subgroup (?) of (i.e., a Variety-containing subgroup of group of prime power order (?)): it contains any subgroup of
that is in the subvariety of the variety of groups generated by
.
In particular, is a Subhomomorph-containing subgroup (?) and a Subisomorph-containing subgroup (?) of
.
Related facts
- Subisomorph-containing implies omega subgroup in group of prime power order
- Omega subgroups not are subisomorph-containing: In particular, the analogous statements break down for groups that are not regular.
Facts used
Proof
By fact (1), is precisely the subset of
comprising the elements of
whose order divides
.
Let be the variety of all groups in which the order of every element divides
. Note that this is a variety and it contains
. Thus, the subvariety generated by
is contained in
. Further, any element of
that is a subgroup of
is contained in
. Thus,
contains all subgroups of
in the subvariety it generates.