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