P-constrained group: Difference between revisions

Revision as of 17:38, 6 March 2009

The article defines a property of groups, where the definition may be in terms of a particular prime that serves as parameter
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Definition

Let $G$ be a finite group and $p$ be a prime number. We say that $G$ is $p$ -constrained if the following is true for one (and hence, any) $p$ -Sylow subgroup of $G$ :

$C_{G} (P \cap O_{p^{'}, p} (G)) \leq O_{p^{'}, p} (G)$ .

Here, $C_{G} (P)$ denotes the centralizer of $P$ in $G$ . $O_{p^{'}, p}$ is the second member of the lower pi-series for $π = {p}$ .

Relation with other properties

Stronger properties

Strongly p-solvable group
p-solvable group: For proof of the implication, refer p-solvable implies p-constrained and for proof of its strictness (i.e. the reverse implication being false) refer p-constrained not implies p-solvable.

Metaproperties

Subgroups

This group property is not subgroup-closed, viz., we can have a group satisfying the property, with a subgroup not satisfying the property

A subgroup of a $p$ -constrained group need not be a $p$ -constrained group. For full proof, refer: p-constrained is not subgroup-closed

Quotients

This group property is not quotient-closed, viz., we could have a group with the property and a quotient group of that group that does not have the property

A quotient of a $p$ -constrained group need not be a $p$ -constrained group. For full proof, refer: p-constrained is not quotient-closed

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 Let <math>G</math> be a [[finite group]] and <math>p</math> be a prime number. We say that <math>G</math> is <math>p</math>-constrained if the following is true for one (and hence, any) <math>p</math>-Sylow subgroup of <math>G</math>:
-<math>C_G(P) \le O_{p',p}(G)</math>.
+<math>C_G(P \cap O_{p',p}(G)) \le O_{p',p}(G)</math>.
 Here, <math>C_G(P)</math> denotes the [[defining ingredient::centralizer]] of <math>P</math> in <math>G</math>. <math>O_{p',p}</math> is the second member of the [[defining ingredient::lower pi-series]] for <math>\pi = \{ p \}</math>.