Thompson's first normal p-complement theorem
From Groupprops
This article gives the statement, and possibly proof, of a normal p-complement theorem: necessary and/or sufficient conditions for the existence of a Normal p-complement (?). In other words, it gives necessary and/or sufficient conditions for a given finite group to be a P-nilpotent group (?) for some prime number .
View other normal p-complement theorems
History
This theorem was proved by Thompson as part of his Ph.D. thesis at the University of Chicago, and was used as a tool to prove the Frobenius conjecture.
Definition
Let be an odd prime number, and a finite group, with -Sylow subgroup . Suppose is a subgroup of the automorphism group of , such that is -invariant. Then suppose the following holds:
"For every -invariant normal subgroup of , the elements of order relatively prime to which normalize , also centralize "
Then is a retract, i.e. possesses a normal p-complement.
Related results
The result does not hold for . Further information: Thompson's normal p-complement theorem fails at the prime two
References
- Normal p-complements for finite groups by John G. Thompson, Math. Zeitschr. 72, 332--354 (1960)
- Finite groups with fixed-point-free automorphisms of prime order by John G. Thompson, Proc. Nat. Acad. Sci. 45, 578-581 (1959)