Čunihin’s theorem

Theorem 1 (Čunihin).

Let G be a finite, π-separable group (http://planetmath.org/Seperable), for some set π of primes. Then

  • any π-subgroupMathworldPlanetmathPlanetmath (http://planetmath.org/PiGroupsAndPiGroups) is contained in a Hall π-subgroup (http://planetmath.org/HallPiSubgroup), and

  • any two Hall π-subgroups are conjugate of one another


  1. 1.

    For π={p}, this essentially reduces to the Sylow theoremsMathworldPlanetmath (with unnecessary hypotheses).

  2. 2.

    If G is solvable, it is π-separable for all π, so such subgroups exist for all π. This result is often called Hall’s theorem. There is another Hall’s theorem, which is similar to this one, can be be found here (http://planetmath.org/HallsTheorem2).


  • 1 Derek J.S. Robinson. A Course in the Theory of Groups, second edition. Springer (1995)
Title Čunihin’s theorem
Canonical name vCunihinsTheorem
Date of creation 2013-03-22 13:17:54
Last modified on 2013-03-22 13:17:54
Owner mathcam (2727)
Last modified by mathcam (2727)
Numerical id 11
Author mathcam (2727)
Entry type Theorem
Classification msc 20D10
Related topic SylowPSubgroups
Defines Hall’s theorem