P-constrained group: Difference between revisions

Latest revision as of 22:18, 6 December 2011

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

Definition for a general finite group

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 $π = {p}$ .

Definition for a p'-core-free finite group

This is the same as the previous definition, restricted to p'-core-free groups.

Let $G$ be a finite group and $p$ be a prime number. Suppose further that the p'-core of $G$ is trivial, i.e., $O_{p^{'}} (G)$ is the trivial group. Equivalently, every nontrivial normal subgroup of $G$ has order divisible by $p$ . Then, we say that $G$ is $p$ -constrained if its p-core is a self-centralizing subgroup, i.e.,:

$C_{G} (O_{p} (G)) \leq O_{p} (G)$

Equivalence of definitions and its significance

Further information: equivalence of definitions of p-constrained group

It turns out that, from the above definitions:

$G$ is $p$ -constrained $⟺$ $G / O_{p^{'}} (G)$ is $p$ -constrained.

This allows us to define $p$ -constraint for arbitrary finite groups in terms of $p$ -constraint for $p^{'}$ -core-free finite groups.

Relation with other properties

Stronger properties

Property	Proof of implication	Proof of strictness (reverse implication failure)
strongly p-solvable group
p-solvable group	p-solvable implies p-constrained	p-constrained not implies p-solvable
finite solvable group	(via p-solvable)	(via p-solvable)
p-nilpotent group	(via p-solvable)	(via p-solvable)
finite nilpotent group	(via finite solvable)	(via finite solvable)

Incomparable properties

Property	Meaning	Proof of one non-implication	Proof of other non-implication
p-stable group		p-constrained not implies p-stable	p-stable not implies p-constrained
group of Glauberman type for a prime		p-constrained not implies Glauberman type	Glauberman type not implies p-constrained

Metaproperties

Metaproperty name	Satisfied?	Proof	Statement with symbols
subgroup-closed group property	No	p-constraint is not subgroup-closed	It is possible to have a finite group $G$ , a subgroup $H$ , and a prime number $p$ such that $G$ is $p$ -constrained and $H$ is not.
quotient-closed group property	No	p-constraint is not quotient-closed	It is possible to have a finite group $G$ , a normal subgroup $N$ , and a prime number $p$ such that $G$ is $p$ -constrained and $G / N$ is not.

@@ Line 2: / Line 2: @@
 ==Definition==
+===Definition for a general finite group===
 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>:
@@ Line 8: / Line 10: @@
 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>.
+===Definition for a p'-core-free finite group===
+This is the same as the previous definition, restricted to p'-core-free groups.
+Let <math>G</math> be a [[finite group]] and <math>p</math> be a prime number. Suppose further that the [[p'-core]] of <math>G</math> is trivial, i.e., <math>O_{p'}(G)</math> is the trivial group. Equivalently, every nontrivial normal subgroup of <math>G</math> has order divisible by <math>p</math>. Then, we say that <math>G</math> is <math>p</math>-constrained if its [[defining ingredient::p-core]] is a [[defining ingredient::self-centralizing subgroup]], i.e.,:
+<math>\! C_G(O_p(G)) \le O_p(G)</math>
+===Equivalence of definitions and its significance===
+{{further|[[equivalence of definitions of p-constrained group]]}}
+It turns out that, from the above definitions:
+<math>G</math> is <math>p</math>-constrained <math>\iff</math> <math>G/O_{p'}(G)</math> is <math>p</math>-constrained.
+This allows us to define <math>p</math>-constraint for arbitrary finite groups in terms of <math>p</math>-constraint for <math>p'</math>-core-free finite groups.
 ==Relation with other properties==
@@ Line 26: / Line 46: @@
 | [[Weaker than::finite nilpotent group]] || || (via finite solvable) || (via finite solvable) ||
 |}
-===Weaker properties===
 ===Incomparable properties===
@@ Line 41: / Line 59: @@
 ==Metaproperties==
-{{not S-closed}}
+{| class="sortable" border="1"
+! Metaproperty name !! Satisfied? !! Proof !! Statement with symbols
-A subgroup of a <math>p</math>-constrained group need not be a <math>p</math>-constrained group. {{proofat|[[p-constrained is not subgroup-closed]]}}
+|-
+| [[dissatisfies metaproperty::subgroup-closed group property]] || No || [[p-constraint is not subgroup-closed]] || It is possible to have a finite group <math>G</math>, a subgroup <math>H</math>, and a prime number <math>p</math> such that <math>G</math> is <math>p</math>-constrained and <math>H</math> is not.
-{{not Q-closed}}
+|-
+| [[dissatisfies metaproperty::quotient-closed group property]] || No || [[p-constraint is not quotient-closed]] || It is possible to have a finite group <math>G</math>, a normal subgroup <math>N</math>, and a prime number <math>p</math> such that <math>G</math> is <math>p</math>-constrained and <math>G/N</math> is not.
-A quotient of a <math>p</math>-constrained group need not be a <math>p</math>-constrained group. {{proofat|[[p-constrained is not quotient-closed]]}}
+|}