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 $$\pi'$$ the set of primes not in $$\pi$$.

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

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

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 $$\pi'$$. Analogously, we define $$O_{\pi'}(H)$$ as the unique largest normal subgroup of $$H$$ such that all prime factors of its order are from $$\pi'$$.
 * We inductively define $$O_{\pi',\pi,\dots,\pi,\pi'}(G)$$ as the group containing $$O_{\pi',\pi,\dots,\pi}(G)$$ such that the quotient $$O_{\pi',\pi,\dots,\pi,\pi'}(G)/O_{\pi',\pi,\dots,\pi}(G)$$ equals $$O_{\pi}(G/O_{\pi',\pi,\dots,\pi}(G))$$. Similarly, we inductively define $$O_{\pi',\pi,\dots,\pi',\pi}(G)$$ as the group containing $$O_{\pi',\pi,\dots,\pi'}(G)$$ such that the quotient $$O_{\pi',\pi,\dots,\pi',\pi}(G)/O_{\pi',\pi,\dots,\pi'}(G)$$ equals $$O_{\pi}(G/O_{\pi',\pi,\dots,\pi'}(G))$$.

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

The adjective lower signifies that we start with $$O_{\pi'}$$. 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 $$\pi'$$-series of the same group. Similarly, the upper $$\pi$$-series coincides with the lower $$\pi'$$-series.
 * If the lower $$\pi$$-series terminates in the whole group, we say that the group is a pi-separable group.