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

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).