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
Statement
Suppose is a finite group,
is a subgroup of
, and
is a set of primes such that:
- 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.
Facts used
Proof
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.