Normal Hall implies permutably complemented
This article gives the statement and possibly, proof, of an implication relation between two subgroup properties. That is, it states that every subgroup satisfying the first subgroup property (i.e., normal Hall subgroup) must also satisfy the second subgroup property (i.e., permutably complemented subgroup)
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This result is sometimes called Schur's theorem, and is considered a part of the Schur-Zassenhaus theorem, which also asserts that any two permutable complements to a normal Hall subgroup are conjugate.
Statement with symbols
Suppose is a normal Hall subgroup of a group . Then there exists a subgroup such that is trivial and . Note that this makes a Hall subgroup as well.