---
tag: KPWW9X14V2
kind: concept
name: Hall subgroup
msc: [20D20]
specializes: [0X1JXT47GA]
sources:
  - url: https://en.wikipedia.org/wiki/Hall_subgroup
same_as:
  wikidata: Q1571916
  wikipedia: Hall subgroup
---

## Summary

A Hall subgroup of a finite group is a [[subgroup|0X1JXT47GA]] whose order is [[coprime|0VZ48N083F]] to its [[index|0DTXBQPA2S]].
Hall subgroups generalise [[Sylow subgroups|0053CQE9Y2]] from one prime to a set of primes.
In a [[solvable group|0FXJ7YFMMX]] they always exist and behave like Sylow subgroups, by [[Hall's theorem|8Z8PE69SJS]]; in other groups they may fail to exist.
They are named after Philip Hall.

## Explanations

### L1 human

A Sylow subgroup collects all the factors of one prime in the order of a group.
A Hall subgroup does the same for a chosen set of primes.
For example, the group $A_5$ has order $60 = 2^2 \cdot 3 \cdot 5$.
Its subgroup $A_4$ has order $12 = 2^2 \cdot 3$ and index $5$, and $12$ and $5$ have no common factor, so $A_4$ is a Hall subgroup for the primes $2$ and $3$.
But $A_5$ has no subgroup of order $15$, so there is no Hall subgroup for the primes $3$ and $5$.