Conjugacy-closed abelian Hall implies retract
Suppose is a finite group and is an abelian conjugacy-closed Hall subgroup of . Then, is a retract of . In other words, possesses a normal complement in : a normal subgroup of such that is trivial and .
- Conjugacy-closed abelian Sylow implies retract
- Conjugacy-closed and Sylow implies retract
- Conjugacy-closed nilpotent Hall implies retract
- Conjugacy-closed and Hall not implies retract
- Burnside's normal p-complement theorem
- Thompson's normal p-complement theorem
- Conjugacy-closed and Hall implies commutator subgroup equals intersection with whole commutator subgroup (this is in turn an immediate corollary of the analogue of focal subgroup theorem for Hall subgroups.
Given: A finite group and an abelian conjugacy-closed Hall subgroup of .
To prove: has a normal complement in .
- is trivial: This follows from fact (1), and the given fact that is abelian.
- has a normal complement: Consider the quotient by , and denote the image of as . Since , the abelianization of , is an Abelian group, and is a Sylow subgroup, there exists a normal complement . Let be the inverse image of under the quotient map. Clearly, , since it contains and its image is the whole of . Further, is trivial, because does not intersect , and the images and intersect trivially. Thus, is the required normal complement.