Hall implies join of Sylow subgroups
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., Hall subgroup) must also satisfy the second subgroup property (i.e., join of Sylow subgroups)
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Given: A finite group , a Hall subgroup .
To prove: is a join of Sylow subgroups.
Proof: Let be the set of prime divisors of the order of . For each , let be a -Sylow subgroup of . Such a exists by fact (1), and is also Sylow in by fact (2).
Now, the join of the s is contained in , because each is contained in . On the other hand, the order of the join of the s must be a multiple of the order of each by Lagrange's theorem, and hence it must be a multiple of their lcm. But the lcm of the orders of the s is the order of , forcing the join of the s to equal .