Pi-separable group

Definition

Let $G$ be a finite group and $\pi$ be a set of primes (we may throw out all the members of $\pi$ that are not divisors of the order of $G$ -- these have no effect). We have the following equivalent formulations for saying that $G$ is a $\pi$-separable group:

No.	Shorthand	$G$ is $\pi$-separable if ...
1	existence of pi-series	there exists a pi-series for $G$, i.e., a subnormal series for $G$ of the form $\{e\}=H_{0}\leq H_{1}\leq \dots H_{n}=G$ where each quotient $H_{i+1}/H_{i}$ is either a $\pi$-group or $\pi '$-group.
2	existence of normal pi-series	there exists a pi-series for $G$ that is a normal series, i.e., all members of the series are normal subgroups of $G$.
3	existence of characteristic pi-series	there exists a pi-series for $G$ that is a characteristic series, i.e., all members of the series are characteristic subgroups of $G$.
4	upper pi-series	the upper pi-series of $G$ terminates in $G$, where the upper $\pi$-series is the series $\{e\}\leq O_{\pi }(G)\leq O_{\pi ,\pi '}(G)\leq O_{\pi ,\pi ',\pi }(G)\leq \dots$.
5	lower pi-series	the lower pi-series of $G$ terminates in $G$, where the lower $\pi$-series is the series $\{e\}\leq O_{\pi '}(G)\leq O_{\pi ',\pi }(G)\leq O_{\pi ',\pi ,\pi '}(G)\leq \dots$.
6	chief series, chief factors	any chief series of the group is a pi-series, i.e., all the chief factors are either $\pi$-groups or $\pi '$-groups.
7	composition series, composition factors	any composition series of the group is a pi-series, i.e., all the composition factors are either $\pi$-groups or $\pi '$-groups.
8	non-abelian composition factors	all the non-abelian composition factors (i.e., all the simple non-abelian groups occurring in a composition series) are either $\pi$-groups or $\pi '$-groups.

$G$ is $\pi$-separable if and only if it is $\pi '$-separable, where $\pi '$ is the complement of $\pi$ in the set of prime divisors of the order of $G$.

The pi-length of $G$ is defined as the half-length of the lower pi-series, i.e., the number of successive quotients of the lower pi-series that are $\pi$-groups.