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

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 $G$:

$C_G(P\cap O_{p^{\prime },p}(G))\le O_{p^{\prime },p}(G)$.

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

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.

Incomparable properties

• p-stable group
• group of Glauberman type for a prime: For full proof, refer: p-constrained not implies Glauberman type, Glauberman type not implies p-constrained

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