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
The following are equivalent for a -Sylow subgroup of a finite group :
- is a normal subgroup of and possesses a normal -complement.
- is a direct factor of .
- is a central factor of .
- is a normal in and is conjugacy-closed in : any two elements of that are conjugate in are conjugate in .
Further information: conjugacy-closed subgroup
Conjugacy-closed normal subgroup
Further information: conjugacy-closed normal subgroup
A subgroup of a group is termed a conjugacy-closed normal subgroup if it is both conjugacy-closed and normal in . Equivalently, is conjugacy-closed normal in if every inner automorphism of restricts to a class-preserving automorphism of : an automorphism that preserves conjugacy classes.
Further information: central factor
Further information: direct factor
- Direct factor implies central factor
- Central factor implies conjugacy-closed normal
- Class-preserving automorphism group of finite p-group is p-group
- Hall and central factor implies direct factor
(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 on by conjugation. Since is conjugacy-closed and normal, every inner automorphism of restricts to a class-preserving automorphism of .
By fact (3), the group of class-preserving automorphisms of is a -group, so the image of the homomorphism given by the action is a -group. Hence, the kernel of the homomorphism, namely , must have index a power of . In particular, by order considerations, , and so is a central factor.
Proof of (3) implies (2)
This follows directly from the stated fact (4).