---
tag: QRWZKTMPPM
kind: statement
role: theorem
name: A converse to Hall's theorem
sources:
  - wikipedia: en:1735335@r1330716647
    locator: Hall subgroup, section A converse to Hall's theorem
    note: adopted
    mention: wikipedia://en/1735335#N875
    sections: [Statement]
inventory: [F3CTD40Q88]
---

## Statement

Any finite group that has a Hall *π*-subgroup for every set of primes *π* is solvable. This is a generalization of [[Burnside's theorem|0FMAQSB1SC]] that any group whose order is of the form *p<sup>a</sup>q<sup>b</sup>* for primes *p* and *q* is solvable, because Sylow's theorem implies that all Hall subgroups exist. This does not (at present) give another proof of Burnside's theorem, because Burnside's theorem is used to prove this [[converse|09A3N7ZN5G]].