Omega subgroups of group of prime power order
Contents
Definition
Suppose is a finite
-group, i.e. a group of prime power order where the prime is
. Then, we define:
In other words, it is the subgroup generated by all elements whose order divides .
If the exponent of is
, then
. However, there may exist smaller
for which
.
The -subgroups form an ascending chain of subgroups:
The -subgroups may also be studied for a (possibly infinite) p-group. Since every element in a p-group, by definition, has order a power of
, the union of the
, for all finite
, is the whole group
. It may still happen that
for some finite
.
Subgroup properties satisfied
All the are clearly characteristic subgroups, and in fact, they're all fully characteristic subgroups: any endomorphism of
sends each
to within
. Even more strongly, all the
s are homomorph-containing subgroups, and for
a finite
-group, they are thus also isomorph-free subgroups.
Further information: Omega subgroups are homomorph-containing
Subgroup-defining function properties
Monotonicity
This subgroup-defining function is monotone, viz the image of any subgroup under this function is contained in the image of the whole group
If is a subgroup, then
.
Idempotence
This subgroup-defining function is idempotent. In other words, applying this twice to a given group has the same effect as applying it once
Applying twice is equivalent to applying it once. In other words, for any
,
.