Pi-separable group: Difference between revisions

Latest revision as of 17:00, 11 September 2011

Template:Prime set-parametrized group property

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.

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 |-
 | 5 || lower pi-series || the [[defining ingredient::lower pi-series]] of <math>G</math> terminates in <math>G</math>, where the lower <math>\pi</math>-series is the series <math>\{ e \} \le O_{\pi'}(G) \le O_{\pi',\pi}(G) \le O_{\pi',\pi,\pi'}(G) \le \dots</math>.
+|-
+| 6 || chief series, chief factors || any [[defining ingredient::chief series]] of the group is a [[pi-series]], i.e., all the [[chief factor]]s are either <math>\pi</math>-groups or <math>\pi'</math>-groups.
+|-
+| 7 || composition series, composition factors || any [[defining ingredient::composition series]] of the group is a [[pi-series]], i.e., all the [[composition factor]]s are either <math>\pi</math>-groups or <math>\pi'</math>-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 <math>\pi</math>-groups or <math>\pi'</math>-groups.
 |}