# 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 |

#### Comments

#### References

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

**How to Cite This Entry:**

Pi-solvable group.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Pi-solvable_group&oldid=48177