Maximal among abelian normal subgroups of Sylow subgroup implies direct factor of centralizer
From Groupprops
Statement
Suppose is a finite group and is a prime number. Suppose is a -Sylow subgroup of , and is Maximal among abelian normal subgroups (?) in . Then, is a Direct factor (?) in .
Related facts
Category:Normal p-complement theorems lists many theorems of a related nature.
Other related facts:
- Hall and central factor implies direct factor
- Pi-separable and pi'-core-free implies pi-core is self-centralizing
Facts used
- Maximal among abelian normal implies self-centralizing in nilpotent
- Product formula
- Burnside's normal p-complement theorem
Proof
Given: A finite group , a prime , a -Sylow subgroup of . A subgroup that is maximal among abelian normal subgroups of .
To prove: is normal in . Further, there exists a subgroup of such that and is trivial.
Proof: Clearly, , hence normal in .
- : This follows from fact (1).
- is a group: Since normalizes , also normalizes . Hence, is a group.
- is a Sylow subgroup of : Consider the intersection . By the product formula (fact (2)), . The right side is relatively prime to , hence so is the left side. By step (1), , so is coprime to . Thus, is a -Sylow subgroup of .
- has a normal complement, say , in : By definition, is in the center of , so it is in the center of its normalizer in , so by fact (3), has a normal complement.
(4) completes the proof.
References
Textbook references
- Finite Groups by Daniel Gorenstein, ISBN 0821843427, Page 259, Theorem 6.5, Chapter 7 (Fusion, transfer and p-factor groups) ,Section 7.6 (elementary applications), ^{More info}