Hall satisfies intermediate subgroup condition
This article gives the statement, and possibly proof, of a subgroup property (i.e., Hall subgroup) satisfying a subgroup metaproperty (i.e., intermediate subgroup condition)
View all subgroup metaproperty satisfactions | View all subgroup metaproperty dissatisfactions |Get help on looking up metaproperty (dis)satisfactions for subgroup properties
Get more facts about Hall subgroup |Get facts that use property satisfaction of Hall subgroup | Get facts that use property satisfaction of Hall subgroup|Get more facts about intermediate subgroup condition
- Sylow satisfies intermediate subgroup condition: Essentially, the same proof.
Given: A finite group , a Hall subgroup , and a subgroup of containing .
To prove: is a Hall subgroup of .
Proof: Note that by the multiplicativity of index:
Thus, the index divides the index . In particular, if and are relatively prime, so are and .