Normal P-complement

In mathematical group theory, a normal p-complement of a finite group for a prime ''p'' is a normal subgroup of order coprime to ''p'' and index a power of ''p''. In other words the group is a semidirect product of the normal ''p''-complement and any Sylow ''p''-subgroup. A group is called p-nilpotent if it has a normal ''p''-complement.

Cayley normal 2-complement theorem

Cayley showed that if the Sylow 2-subgroup of a group ''G'' is cyclic then the group has a normal 2-complement, which shows that the Sylow 2-subgroup of a simple group of even order cannot be cyclic.

Burnside normal p-complement theorem

showed that if a Sylow ''p''-subgroup of a group ''G'' is in the center of its normalizer then ''G'' has a normal ''p''-complement. This implies that if ''p'' is the smallest prime dividing the order of a group ''G'' and the Sylow ''p''-subgroup is cyclic, then ''G'' has a normal ''p''-complement.

Frobenius normal p-complement theorem

The Frobenius normal ''p''-complement theorem is a strengthening of the Burnside normal ''p''-complement theorem, that states that if the normalizer of every non-trivial subgroup of a Sylow ''p''-subgroup of ''G'' has a normal ''p''-complement, then so does ''G''. More precisely, the following conditions are equivalent:
*''G'' has a normal ''p''-complement
*The normalizer of every non-trivial ''p''-subgroup has a normal ''p''-complement
*For every ''p''-subgroup ''Q'', the group N_''G''(''Q'')/C_''G''(''Q'') is a ''p''-group.

Thompson normal p-complement theorem

The Frobenius normal ''p''-complement theorem shows that if every normalizer of a non-trivial subgroup of a Sylow ''p''-subgroup has a normal ''p''-complement then so does ''G''. For applications it is often useful to have a stronger version where instead of using all non-trivial subgroups of a Sylow ''p''-subgroup, one uses only the non-trivial characteristic subgroups. For odd primes ''p'' Thompson found such a strengthened criterion: in fact he did not need all characteristic subgroups, but only two special ones.

showed that if ''p'' is an odd prime and the groups N(J(''P'')) and C(Z(''P'')) both have normal ''p''-complements for a Sylow P-subgroup of ''G'', then ''G'' has a normal ''p''-complement.

In particular if the normalizer of every nontrivial characteristic subgroup of ''P'' has a normal ''p''-complement, then so does ''G''. This consequence is sufficient for many applications.

The result fails for ''p'' = 2 as the simple group PSL₂(F₇) of order 168 is a counterexample.

gave a weaker version of this theorem.

Glauberman normal p-complement theorem

Thompson's normal ''p''-complement theorem used conditions on two particular characteristic subgroups of a Sylow ''p''-subgroup. Glauberman improved this further by showing that one only needs to use one characteristic subgroup: the center of the Thompson subgroup.

used his ZJ theorem to prove a normal ''p''-complement theorem, that if ''p'' is an odd prime and the normalizer of Z(J(P)) has a normal ''p''-complement, for ''P'' a Sylow ''p''-subgroup of ''G'', then so does ''G''. Here ''Z'' stands for the center of a group and ''J'' for the Thompson subgroup.

The result fails for ''p'' = 2 as the simple group PSL₂(F₇) of order 168 is a counterexample.

References

* Reprinted by Dover 1955
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*{{Citation , last1=Thompson , first1=John G. , author1-link=John G. Thompson , title=Normal p-complements for finite groups , doi=10.1016/0021-8693(64)90006-7 , mr=0167521 , year=1964 , journal=Journal of Algebra , issn=0021-8693 , volume=1 , pages=43–46, doi-access=free

Finite groups