# Normal p-complement

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of a finite group

A normal subgroup such that and , where is a Sylow -subgroup of (see Sylow subgroup). A group has a normal -complement if some Sylow -subgroup of 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 -complement in a group is given by Frobenius' theorem: A group has a normal -complement if and only either for any non-trivial -subgroup of the quotient group is a -group (where is the normalizer and the centralizer of in ) or if for every non-trivial -subgroup of the subgroup has a normal -complement.

#### References

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