# 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)