# P-constrained group

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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 $G$: $C_G(P \cap O_{p',p}(G)) \le O_{p',p}(G)$.

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

### 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.,: $\! C_G(O_p(G)) \le O_p(G)$

### 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 $\iff$ $G/O_{p'}(G)$ 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 Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
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.