Omega subgroups of a p-group

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Definition

Suppose $P$ is a finite $p$ -group, i.e. a group of prime power order where the prime is $p$ . Then, we define:

$\Omega _{j}(P):=\langle x\in P\mid x^{p^{j}}=e\rangle$

In other words, it is the subgroup generated by all elements whose order divides $p^{j}$ .

If the exponent of $P$ is $p^{r}$ , then $\Omega _{r}(P)=P$ . However, there may exist smaller $j$ for which $\Omega _{j}(P)=P$ .

The $\Omega$ -subgroups form an ascending chain of subgroups:

$\{e\}=\Omega _{0}(P)\leq \Omega _{1}(P)\leq \dots \leq \Omega _{r}(P)=P$

The $\Omega$ -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 $p$ , the union of the $\Omega _{j}(P)$ , for all finite $j$ , is the whole group $P$ . It may still happen that $\Omega _{j}(P)=P$ for some finite $j$ .

Subgroup properties satisfied

All the $\Omega _{j}$ are clearly characteristic subgroups, and in fact, they're all fully characteristic subgroups: any endomorphism of $P$ sends each $\Omega _{j}(P)$ to within $\Omega _{j}(P)$ . Even more strongly, all the $\Omega _{j}$ s are homomorph-containing subgroups, and for $P$ a finite $p$ -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 $Q\leq P$ is a subgroup, then $\Omega _{j}(Q)\leq \Omega _{j}(P)$ .

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 $\Omega _{j}$ twice is equivalent to applying it once. In other words, for any $P$ , $\Omega _{j}(\Omega _{j}(P))=\Omega _{j}(P)$ .