P-normal group

The article defines a property of groups, where the definition may be in terms of a particular prime that serves as parameter
View other prime-parametrized group properties | View other group properties

Definition

Suppose $G$ is a finite group and $p$ is a prime number. We say that $G$ is $p$-normal if it satisfies the following equivalent conditions:

1. There exists a $p$-Sylow subgroup $P$ of $G$ such that the center $Z(P)$ is a weakly closed subgroup of $P$ relative to $G$.
2. For every $p$-Sylow subgroup $P$ of $G$, the center $Z(P)$ is a weakly closed subgroup of $P$ relative to $G$.
3. There exists a $p$-Sylow subgroup $P$ of $G$ with center $Z(P)$ such that for any $p$-Sylow subgroup $Q$ of $G$ containing $P$, $Z(P)$ is a normal subgroup of $Q$.
4. For every $p$-Sylow subgroup $P$ of $G$ with center $Z(P)$, it is true that for any $p$-Sylow subgroup $Q$ of $G$ containing $P$, $Z(P)$ is a normal subgroup of $Q$.

Equivalence of definitions

The equivalence between definitions (1)-(2) and between definitions (3)-(4) follows from the fact that Sylow implies order-conjugate: any two $p$-Sylow subgroups are conjugate, and the conjugating automorphism preserves all properties including weak closure. The equivalence between (1) and (3) follows from the fact that characteristic subgroup of Sylow subgroup is weakly closed iff it is normal in all Sylow subgroups containing it.