Hall satisfies permuting transfer condition
This article gives the statement, and possibly proof, of a subgroup property (i.e., Hall subgroup) satisfying a subgroup metaproperty (i.e., permuting transfer condition)
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Statement with symbols
Further information: Hall subgroup
- Index is multiplicative
- Lagrange's theorem
- Product formula: This states that if and are subgroups of , we have:
Given: A finite group , a Hall subgroup of , a subgroup of such that .
To prove: is Hall in .
Proof: Rearranging the product formula (fact (3)) yields:
By Lagrange's theorem (fact (2)), and noting that is a subgroup of , we get:
By fact (1), we have:
Thus, we get:
In particular, divides . By Lagrange's theorem, we have that divides . Since and are relatively prime, we obtain that and are relatively prime. Thus, is a Hall subgroup of .