Equivalence of definitions of Sylow direct factor

This article gives a proof/explanation of the equivalence of multiple definitions for the term Sylow direct factor
View a complete list of pages giving proofs of equivalence of definitions

Statement

The following are equivalent for a $p$-Sylow subgroup $P$ of a finite group $G$:

1. $P$ is a normal subgroup of $G$ and possesses a normal $p$-complement.
2. $P$ is a direct factor of $G$.
3. $P$ is a central factor of $G$.
4. $P$ is a normal in $G$ and is conjugacy-closed in $G$: any two elements of $P$ that are conjugate in $G$ are conjugate in $P$.

Related facts

• Conjugacy-closed Abelian Sylow implies retract
• Conjugacy-closed and Sylow implies retract

Definitions used

Conjugacy-closed subgroup

Further information: conjugacy-closed subgroup

Conjugacy-closed normal subgroup

Further information: conjugacy-closed normal subgroup

A subgroup $H$ of a group $G$ is termed a conjugacy-closed normal subgroup if it is both conjugacy-closed and normal in $G$. Equivalently, $H$ is conjugacy-closed normal in $G$ if every inner automorphism of $G$ restricts to a class-preserving automorphism of $H$: an automorphism that preserves conjugacy classes.

Central factor

Further information: central factor

Direct factor

Further information: direct factor

Facts used

1. Direct factor implies central factor
2. Central factor implies conjugacy-closed normal
3. Class-preserving automorphism group of finite p-group is p-group
4. Hall and central factor implies direct factor

Proof

(1) and (2) are equivalent by the definition of direct factor. (2) implies (3) by fact (1) and (3) implies (4) by fact (2). Thus, it suffices to show that (4) implies (3) and (3) implies (2).

Proof of (4) implies (3)

Consider the action of $G$ on $P$ by conjugation. Since $P$ is conjugacy-closed and normal, every inner automorphism of $G$ restricts to a class-preserving automorphism of $P$.

By fact (3), the group of class-preserving automorphisms of $P$ is a $p$-group, so the image of the homomorphism $G\to \operatorname {Aut} (P)$ given by the action is a $p$-group. Hence, the kernel of the homomorphism, namely $C_{G}(P)$, must have index a power of $p$. In particular, by order considerations, $PC_{G}(P)=G$, and so $P$ is a central factor.

Proof of (3) implies (2)

This follows directly from the stated fact (4).