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)