# Pi-solvable group

A generalization of the concept of a solvable group. Let 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 or coincides with a certain prime in , is called a -solvable group. The basic properties of -solvable groups are similar to the properties of solvable groups. A -solvable group is a -solvable group for any ; the subgroups, quotient groups and extensions of a -solvable group by a -solvable group are also -solvable groups. In a -solvable group every -subgroup (that is, a subgroup all prime factors of the order of which belong to ) is contained in some Hall -subgroup (a Hall -subgroup is one with index in the group not divisible by any prime in ) and every -subgroup (where is the complement of in the set of all prime numbers) is contained in some Hall -subgroup; all Hall -subgroups and also all Hall -subgroups are conjugate in ; the index of a maximal subgroup of the group is either not divisible by any number in or is a power of one of the numbers of the set (see [1]). The number of Hall -subgroup in is equal to , where () for every which divides the order of , and, moreover, divides the order of one of the chief factors of (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 |

#### Comments

#### References

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

**How to Cite This Entry:**

Pi-solvable group. S.P. Strunkov (originator),

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Pi-solvable_group&oldid=15840