# Čunihin’s theorem

###### Theorem 1 (Čunihin).

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

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

• any two Hall $\pi$-subgroups are conjugate of one another

Remarks

1. 1.

For $\pi=\{p\}$, this essentially reduces to the Sylow theorems (with unnecessary hypotheses).

2. 2.

If $G$ is solvable, it is $\pi$-separable for all $\pi$, so such subgroups exist for all $\pi$. 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).

## References

• 1 Derek J.S. Robinson. A Course in the Theory of Groups, second edition. Springer (1995)
Title Čunihin’s theorem vCunihinsTheorem 2013-03-22 13:17:54 2013-03-22 13:17:54 mathcam (2727) mathcam (2727) 11 mathcam (2727) Theorem msc 20D10 SylowPSubgroups Hall’s theorem