Hall subgroup
In group theory , a branch of algebra , a Hall subgroup is understood to be a subgroup of a finite group , the thickness of which is coprime to its index .
They are named after the British mathematician Philip Hall .
Formal definition
Be a finite group .
is called hallsch in if and only if and are coprime .
Note that this definition only makes sense for finite groups, because the index and the cardinality of a subgroup of an infinite group cannot both be finite.
Examples
- Each Sylow group is echoing in the respective group
- Each group is echoing in itself
- The Frobenius complement of a Frobenius group is echoing in the group
- The alternating group of degree is reverberant in the symmetrical group of degree if and only if
meaning
Philip Hall showed that for every finite solvable group and a set of prime numbers we have:
- (1) has Hallsche subgroups
- (2) Two such subgroups are conjugated
- (3) Each subgroup of is in a hall's subgroup of containing
A subgroup of is a group whose order contains all the numbers from .
Conversely, every finite group that has a corresponding Hall subgroup for every set of prime numbers can be resolved.