Burnside's 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

Name

This result is termed Burnside's normal p-complement theorem and is also sometimes termed Burnside's transfer theorem.

Statement

Statement with symbols

Suppose p is a prime, G is a finite group, and P is a p-Sylow subgroup (?) of G. Further, suppose P is a Central subgroup of normalizer (?): if H = N_G(P) is its normalizer, and Z(H) is the center of H, then P \le Z(H).

Then, P is a Retract (?) of G, i.e., there exists a normal p-complement in G: a normal subgroup N such that NP = G and N \cap P is trivial.

Related facts

Facts used

  1. Center of Sylow sugbroup is conjugacy-determined in normalizer: If P is a Sylow subgroup of G, then two elements of Z(P) are conjugate in G if and only if they are conjugate in N_G(P).
  2. Conjugacy-closed abelian Sylow implies retract: If P is an Sylow subgroup of G such that no two distinct elements of P are conjugate in G, then P is a retract of G. (Note that the proof of this relies in turn on the focal subgroup theorem).
  3. Grün's first theorem on the focal subgroup

Proof

Proof using a rather weak fusion result

Given: G a finite group, P a p-Sylow subgroup such that P \le Z(N_G(P)).

To prove: P is a retract of G: it possesses a normal p-complement.

Proof:

  1. No two distinct elements of P are conjugate in G: Since P is Abelian, Z(P) = P, so fact (1) tells us that two elements of P are conjugate in G if and only if they are conjugate in N_G(P). Since P \le Z(N_G(P)), no two distinct elements of P are conjugate in N_G(P), and hence no two distinct elements of P are conjugate in G.
  2. P has a normal complement: This follows from fact (2), since the previous step shows that the conditions for it are satisfied.

Proof using a stronger fusion result

This proof uses fact (3).