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 . ${\displaystyle G}$ ${\displaystyle U\leq G}$

${\displaystyle U}$is called hallsch in if and only if and are coprime . ${\displaystyle G}$ ${\displaystyle |U|}$ ${\displaystyle (G:U)}$

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 ${\displaystyle n}$ ${\displaystyle n}$ ${\displaystyle n\leq 3}$

meaning

Philip Hall showed that for every finite solvable group and a set of prime numbers we have: ${\displaystyle G}$ ${\displaystyle \pi }$

(1) has Hallsche subgroups ${\displaystyle G}$ ${\displaystyle \pi }$

(2) Two such subgroups are conjugated

(3) Each subgroup of is in a hall's subgroup of containing ${\displaystyle \pi }$ ${\displaystyle G}$ ${\displaystyle \pi }$ ${\displaystyle G}$

A subgroup of is a group whose order contains all the numbers from . ${\displaystyle \pi }$ ${\displaystyle G}$ ${\displaystyle \pi }$

Conversely, every finite group that has a corresponding Hall subgroup for every set of prime numbers can be resolved. ${\displaystyle \pi }$