P-stable group: Difference between revisions

Revision as of 01:29, 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 a $p$ -stable group if $O_{p} (G)$ is nontrivial, and $G$ satisfies the following:

Suppose $P$ is a $p$ -subgroup of $G$ such that $O_{p^{'}} (G) P$ is normal in $G$ . Then, if $A$ is a $p$ -subgroup of $N_{G} (P)$ with the property that $[[P, A], A]$ is trivial, we have:

$A C_{G} (P) / C_{G} (P) \leq O_{p} (N_{G} (P) / C_{G} (P))$ .

Relation with other properties

Stronger properties

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

References

Textbook references

Finite Groups by Daniel Gorenstein, ISBN 0821843427, Page 268, Chapter 8 (p-constrained and p-stable groups), Section 8.1 (p-constraint and p-stability), ^{More info}

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 ==Definition==
-Let <math>G</math> be a [[finite group]] and <math>p</math> be a prime number. Suppose <math>P</math> is a <math>p</math>-subgroup of <math>G</math> such that <math>O_{p'}(G)P</math> is normal in <math>G</math>. Then, if <math>A</math> is a <math>p</math>-subgroup of <math>N_G(P)</math> with the property that <math>[[P,A],A]</math> is trivial, we have:
+Let <math>G</math> be a [[finite group]] and <math>p</math> be a prime number. We say that <math>G</math> is a <math>p</math>-stable group if <math>O_p(G)</math> is nontrivial, and <math>G</math> satisfies the following:
+Suppose <math>P</math> is a <math>p</math>-subgroup of <math>G</math> such that <math>O_{p'}(G)P</math> is normal in <math>G</math>. Then, if <math>A</math> is a <math>p</math>-subgroup of <math>N_G(P)</math> with the property that <math>[[P,A],A]</math> is trivial, we have:
 <math>AC_G(P)/C_G(P) \le O_p(N_G(P)/C_G(P))</math>.