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. The second half is available at Hall retract implies order-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.
Equivalently, if is a normal -Hall subgroup of , then possesses a -Hall subgroup: a Hall subgroup corresponding to the primes not in .
- Abelian normal Hall implies permutably complemented: This was the original version of the theorem proved by Schur.
- Sylow subgroups exist
- Frattini's argument
- Normal Hall satisfies transfer condition
- Equivalence of definitions of finite nilpotent group
- Center of normal implies normal
We prove this claim by induction on the order of . Rather, we reduce the claim to (1) by induction on the order of .
Given: A group , a normal Hall subgroup of .
To prove: has a permutable complement in .
Proof: Note that for any prime dividing the order of , a -Sylow subgroup of is also -Sylow in . Note also that by fact (2), -Sylow subgroups exist for all primes .
- Case where there is a prime dividing the order of , such that a -Sylow subgroup of is not normal in : Let . by fact (3) (Frattini's argument). Further, by fact (4), is a normal Hall subgroup of , so by induction, there exists a subgroup of that is a permutable complement in to . We claim that is a permutable complement to in :
- is trivial: Since and is trivial, we obtain that is trivial.
- : We have . Also, . Thus, . But as observed earlier, by fact (3), . this forces .
- Case where is non-abelian and, for every prime dividing the order of , a -Sylow subgroup of is normal in : In this case, each of the -Sylow subgroups of is normal in , hence is a direct product of its Sylow subgroups. Thus, is nilpotent. In particular, the center is nontrivial. By fact (6), is normal in . Consider the group . is a normal Hall subgroup of , so by induction, there exists a permutable complement to in . Suppose is the inverse image of under the quotient map . Note that is a proper subgroup of , since is a proper subgroup of by the assumption that is not abelian. and have relatively prime orders, so is a normal Hall subgroup of . Thus, there exists a permutable complement to in .
- Case that is abelian: This is covered by fact (1).