Normal p-complement

of a finite group $ G $

A normal subgroup $ A $ such that $ G = AS $ and $ A \cap S = 1 $, where $ S $ is a Sylow $ p $- subgroup of $ G $( see Sylow subgroup). A group $ G $ has a normal $ p $- complement if some Sylow $ p $- subgroup $ S $ of $ G $ lies in the centre of its normalizer (cf. Normalizer of a subset) (Burnside's theorem). A necessary and sufficient condition for the existence of a normal $ p $- complement in a group $ G $ is given by Frobenius' theorem: A group $ G $ has a normal $ p $- complement if and only either for any non-trivial $ p $- subgroup $ H $ of $ G $ the quotient group $ N _ {G} ( H)/ C _ {G} ( H) $ is a $ p $- group (where $ N _ {G} ( H) $ is the normalizer and $ C _ {G} ( H) $ the centralizer of $ H $ in $ G $) or if for every non-trivial $ p $- subgroup $ H $ of $ G $ the subgroup $ N _ {G} ( H) $ has a normal $ p $- complement.

References

[1]	D. Gorenstein, "Finite groups" , Harper & Row (1968)

Comments

Let $ G $ be a group of order $ n $ and let $ p ^ {e} $ be the highest power of a prime number $ p $ dividing $ n $. A subgroup of $ G $ of index $ p ^ {e} $( and hence of order $ p ^ {-} e n $) is called a $ p $- complement in $ G $. A normal $ p $- complement is a $ p $- complement that is normal. A finite group is solvable if and only if it has a $ p $- complement for every prime number $ p $ dividing its order. Cf. [a1], [a2] for more details; cf. also Hall subgroup.

References

[a1]	M. Hall jr., "The theory of groups" , Macmillan (1959) pp. Sect. 9.3
[a2]	B. Huppert, "Endliche Gruppen" , 1 , Springer (1967) pp. Sect. VI.1