View source for Pi-solvable group

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A generalization of the concept of a [[Solvable group|solvable group]]. Let  $  \pi $
be a certain set of prime numbers. A finite group for which the order of each composition factor either is coprime to any member of  $  \pi $
or coincides with a certain prime in  $  \pi $, 
is called a  $  \pi $-
solvable group. The basic properties of  $  \pi $-
solvable groups are similar to the properties of solvable groups. A  $  \pi $-
solvable group is a  $  \pi _ {1} $-
solvable group for any  $  \pi _ {1} \subset  \pi $; 
the subgroups, quotient groups and extensions of a  $  \pi $-
solvable group by a  $  \pi $-
solvable group are also  $  \pi $-
solvable groups. In a  $  \pi $-
solvable group  $  G $
every  $  \pi $-
subgroup (that is, a subgroup all prime factors of the order of which belong to  $  \pi $) 
is contained in some Hall  $  \pi $-
subgroup (a Hall  $  \pi $-
subgroup is one with index in the group not divisible by any prime in  $  \pi $) 
and every  $  \pi  ^  \prime  $-
subgroup (where  $  \pi  ^  \prime  $
is the complement of  $  \pi $
in the set of all prime numbers) is contained in some Hall  $  \pi  ^  \prime  $-
subgroup; all Hall  $  \pi $-
subgroups and also all Hall  $  \pi  ^  \prime  $-
subgroups are conjugate in  $  G $; 
the index of a maximal subgroup of the group  $  G $
is either not divisible by any number in  $  \pi $
or is a power of one of the numbers of the set  $  \pi $(
see [[#References|[1]]]). The number of Hall  $  \pi $-
subgroup in  $  G $
is equal to  $  \alpha _ {1} \dots \alpha _ {t} $, 
where  $  \alpha _ {i} \equiv 1 $(
$   \mathop{\rm mod}  p _ {i} $) 
for every  $  p _ {i} \in \pi $
which divides the order of  $  G $, 
and, moreover,  $  \alpha _ {i} $
divides the order of one of the chief factors of  $  G $(
see [[#References|[2]]]).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S.A. Chunikhin,   "Subgroups of finite groups" , Wolters-Noordhoff  (1969)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  W. Brauer,   "Zu den Sylowsätzen von Hall und Čunichin"  ''Arch. Math.'' , '''19''' :  3  (1968)  pp. 245–255</TD></TR></table>

====Comments====

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  D.J.S. Robinson,   "A course in the theory of groups" , Springer  (1982)</TD></TR></table>