Hall subgroup

Under a hall between subgroup is understood in group theory, a field of algebra, a subgroup of a finite group whose cardinality is prime to its index.

They are named after the British mathematician Philip Hall.

Formal definition

Let be a finite group.

Ie hallsch in if and only if and are relatively prime.

Note that this definition makes sense only for finite groups, because the index and the thickness not both must finally be a subgroup of an infinite group.

Examples

• Each Sylowgruppe is hallsch in each group
• Each group is hallsch in itself
• The Frobeniuskomplement a Frobenius is hallsch in the group
• The alternating group of degree exactly then hallsch in the symmetric group of degree, if

Importance

Philip Hall has shown that the following holds for every finite solvable group and a set of prime numbers:

This is a subgroup of a group whose order contains all the numbers from.

Conversely, every finite group has a corresponding hallsche subgroup at any set of primes, dissolvable.

• The finite set
• Group Theory