Sylow satisfies intermediate subgroup condition
This article gives the statement, and possibly proof, of a subgroup property (i.e., Sylow subgroup) satisfying a subgroup metaproperty (i.e., intermediate subgroup condition)
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Further information: Sylow subgroup
is termed a -Sylow subgroup if its order is a power of the prime and its index is not a multiple of .
Given: A finite group , a -Sylow subgroup of , a subgroup of containing .
To prove: is a Sylow subgroup (in fact, a -Sylow subgroup) inside .
Proof: Clearly, continues to be a -subgroup inside , because it is a -group. Thus, it suffices to show that the index is relatively prime to . For this, observe that by fact (1):
Thus, is a divisor of . Since is relatively prime to , so is .