Hall is transitive
This article gives the statement, and possibly proof, of a subgroup property (i.e., Hall subgroup) satisfying a subgroup metaproperty (i.e., transitive subgroup property)
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Given: A finite group , subgroups such that is a Hall subgroup of and is a Hall subgroup of .
To prove: is a Hall subgroup of .
Proof: By fact (1), we have:
Now, since is Hall in , the order of is relatively prime to .
By fact (2), the order of divides the order of , and since is a Hall subgroup of ,the order of is relatively prime to . Thus, the order of is relatively prime to .
Thus, the order of is relatively prime to the product , which, by the above equation, equals .