Thompson's first normal p-complement theorem

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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 p.
View other normal p-complement theorems


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.


Let p be an odd prime number, and G a finite group, with p-Sylow subgroup P. Suppose A is a subgroup of the automorphism group of G, such that P is A-invariant. Then suppose the following holds:

"For every A-invariant normal subgroup Q of P, the elements of order relatively prime to p which normalize Q, also centralize Q"

Then P is a retract, i.e. G possesses a normal p-complement.

Related results

The result does not hold for p=2. Further information: Thompson's normal p-complement theorem fails at the prime two


  • 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)