Omega subgroups of group of prime power order
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
If is a subgroup, then .
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 , .