View source for Pi-separable group

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A group which has a normal series such that the order of every factor contains at most one prime from $\pi$ ($\pi$ is a set of prime numbers). The class of $\pi$-separable groups contains the class of $\pi$-solvable groups (cf. [[Pi-solvable group|$\pi$-solvable group]]). For finite $\pi$-separable groups, the $\pi$-Sylow properties (cf. [[Sylow theorems|Sylow theorems]]) have been shown to hold (see [[#References|[1]]]). In fact, for any set $\pi_1\subseteq\pi$, a finite $\pi$-separable group $G$ contains a $\pi_1$-Hall subgroup (cf. also [[Hall subgroup|Hall subgroup]]), and any two $\pi_1$-Hall subgroups are conjugate in $G$. Any $\pi_1$-subgroup of a $\pi$-separable group $G$ is contained in some $\pi_1$-Hall subgroup of $G$ (see [[#References|[2]]]).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S.A. Chunikhin,   "On <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071000/p07100020.png" />-separable groups"  ''Dokl. Akad. Nauk SSSR'' , '''59''' :  3  (1948)  pp. 443–445  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  P. Hall,   "Theorems like Sylow's"  ''Proc. London Math. Soc.'' , '''6''' :  22  (1956)  pp. 286–304</TD></TR></table>



====Comments====
Chunikhin's theorem says that if $k$ is a divisor of the order $n$ of a $\pi$-separable group such that $n=kl$, $(k,l)=1$, and if all prime divisors of $k$ are in $\pi$, then $G$ has a subgroup of order $k$ and all these subgroups are conjugate in $G$. If $\pi$ consists of all prime numbers this becomes Hall's first theorem.

Gol'berg's theorem, [[#References|[a2]]], says that if $G$ is a finite $\pi$-separable group and $\pi_1$ is a subset of $\pi$, then $G$ has a Sylow $\pi_1$-basis (cf. [[Sylow basis|Sylow basis]]) and all these bases are conjugate.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A.G. Kurosh,   "The theory of groups" , '''2''' , Chelsea  (1956)  pp. 195ff  (Translated from Russian)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  P.A. Gol'berg,   "Sylow bases of $\pi$-separable groups"  ''Dokl. Akad. Nauk SSSR'' , '''60'''  (1949)  pp. 615–618  (In Russian)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  S.A. Chunikhin,   "On $\pi$-properties of finite groups"  ''Mat. Sb.'' , '''25'''  (1949)  pp. 321–346  (In Russian)</TD></TR></table>