Pi-dominating pi-subgroup implies pi-Hall
This article gives a proof/explanation of the equivalence of multiple definitions for the term order-dominating Hall subgroup
View a complete list of pages giving proofs of equivalence of definitions
- The set of prime divisors of the order of is in .
- Given any -subgroup of (i.e., any subgroup for which all prime divisors of its order are in ), there exists such that .
Then, is a -Hall subgroup: in particular, its index is relatively prime to , or equivalently, its order is the unique largest -number dividing the order of .
Note that this also shows that a subgroup is -dominating for a set of primes iff it is an order-dominating Hall subgroup.
Given: A finite group , a -dominating subgroup .
To prove: is -Hall.
Proof: It suffices to show that for every , the largest power of dividing the order of also divides the order of .
For this, let be a -Sylow subgroup (existence follows from fact (1)). The order of is the largest power of dividing the order of . By the assumption, some conjugate is in . The order of equals the order of , and by fact (2), this divides the order of . Thus, the order of is a multiple of the largest power of dividing the order of , and this completes the proof.