P-constrained group

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.