Lower pi-series

Definition

Let $G$ be a finite group and $\pi$ be a set of primes (we can, without loss of generality, assume $\pi$ to be a subset of the set of primes dividing the order of $G$, because primes that don't divide the order of $G$ play no role). We denote by ${\pi }^{\prime }$ the set of primes not in $\pi$.

The lower $\pi$-series of $G$ is a series defined as follows:

$\{e\}\le O_{{\pi }^{\prime }}(G)\le O_{{\pi }^{\prime },\pi }(G)\le O_{{\pi }^{\prime },\pi ,{\pi }^{\prime }}(G)\le \dots$

Here is a description of the members:

• For a group $H$, $O_{\pi }(H)$, also called the pi-core of $H$, is the unique largest normal subgroup of $H$ such that all prime factors of its order are from $\pi$, and therefore none from ${\pi }^{\prime }$. Analogously, we define $O_{{\pi }^{\prime }}(H)$ as the unique largest normal subgroup of $H$ such that all prime factors of its order are from ${\pi }^{\prime }$.
• We inductively define $O_{{\pi }^{\prime },\pi ,\dots ,\pi ,{\pi }^{\prime }}(G)$ as the group containing $O_{{\pi }^{\prime },\pi ,\dots ,\pi }(G)$ such that the quotient $O_{{\pi }^{\prime },\pi ,\dots ,\pi ,{\pi }^{\prime }}(G)/O_{{\pi }^{\prime },\pi ,\dots ,\pi }(G)$ equals $O_{\pi }(G/O_{{\pi }^{\prime },\pi ,\dots ,\pi }(G))$. Similarly, we inductively define $O_{{\pi }^{\prime },\pi ,\dots ,{\pi }^{\prime },\pi }(G)$ as the group containing $O_{{\pi }^{\prime },\pi ,\dots ,{\pi }^{\prime }}(G)$ such that the quotient $O_{{\pi }^{\prime },\pi ,\dots ,{\pi }^{\prime },\pi }(G)/O_{{\pi }^{\prime },\pi ,\dots ,{\pi }^{\prime }}(G)$ equals $O_{\pi }(G/O_{{\pi }^{\prime },\pi ,\dots ,{\pi }^{\prime }}(G))$.

In other words, for each successive quotient, we alternate between $O_{\pi }$ and $O_{{\pi }^{\prime }}$.

The adjective lower signifies that we start with $O_{{\pi }^{\prime }}$. If we start with $O_{\pi }$, we get the upper pi-series.

Facts

• The lower $\pi$-series of a finite group coincides with the upper ${\pi }^{\prime }$-series of the same group. Similarly, the upper $\pi$-series coincides with the lower ${\pi }^{\prime }$-series.
• If the lower $\pi$-series terminates in the whole group, we say that the group is a pi-separable group.