# 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
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## Definition

### Short version

Suppose $G$ is a finite group and $p$ is a prime number. We say that $G$ is $p$-normal if the conjugacy functor on $G$ arising from the characteristic p-functor sending a finite p-group to its center is a weakly closed conjugacy functor on $G$.

### Long version

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. Either of these equivalent:
• There exists a $p$-Sylow subgroup $P$ of $G$ such that the center $Z(P)$ is a weakly closed subgroup of $P$ relative to $G$.
• For every $p$-Sylow subgroup $P$ of $G$, the center $Z(P)$ is a weakly closed subgroup of $P$ relative to $G$.
2. Either of these equivalent:
• There exists a $p$-Sylow subgroup $P$ such that, for every $p$-Sylow subgroup $Q$ containing the center $Z(P)$, $Z(P) = Z(Q)$.
• For every $p$-Sylow subgroup $P$, and for every $p$-Sylow subgroup $Q$ containing the center $Z(P)$, $Z(P) = Z(Q)$.
3. Either of these equivalent:
• There exists a $p$-Sylow subgroup $P$ of $G$ with center $Z(P)$ such that for any $p$-Sylow subgroup $Q$ of $G$ containing the center $Z(P)$, $Z(P)$ is a normal subgroup of $Q$.
• 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 the center $Z(P)$, $Z(P)$ is a normal subgroup of $Q$.

### Equivalence of definitions

Further information: equivalence of definitions of weakly closed conjugacy functor

The equivalence between both versions of (1), the equivalence between both versions of (2), and the equivalence between both versions of (3), follow 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 (1) implies (2) implies (3) direction is straightforward. The (3) implies (1) direction follows from the fact that characteristic subgroup of Sylow subgroup is weakly closed iff it is normal in every Sylow subgroup containing it.

## Relation with other properties

### Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
p-nilpotent group there is a p'-Hall subgroup, i.e., a normal p-complement. The $p$-Sylow subgroup is thus a retract. p-nilpotent implies p-normal p-normal not implies p-nilpotent |FULL LIST, MORE INFO
group in which the $p$-Sylow subgroup is abelian