Pi-solvable group

A generalization of the concept of a 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 [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 [2]).

References

[1]	S.A. Chunikhin, "Subgroups of finite groups" , Wolters-Noordhoff (1969) (Translated from Russian)
[2]	W. Brauer, "Zu den Sylowsätzen von Hall und Čunichin" Arch. Math. , 19 : 3 (1968) pp. 245–255

References

[a1]	D.J.S. Robinson, "A course in the theory of groups" , Springer (1982)