# Supersolvable group

From Encyclopedia of Mathematics

*supersoluble group*

A group $G$ with a finite series of normal subgroups (cf. Subgroup series)

$$G=G_1\supseteq G_2\supseteq\dots\supseteq G_{n+1}=E,$$

in which each quotient group $G_{i-1}/G_i$ is cyclic. Every supersolvable group is a polycyclic group. Subgroups and quotient groups of a supersolvable group are also supersolvable, and the commutator subgroup of a supersolvable group is nilpotent. A finite group is supersolvable if and only if all of its maximal subgroups have prime index (Huppert's theorem).

#### Comments

#### References

[a1] | K. Doerk, T. Hawkes, "Finite soluble groups" , de Gruyter (1992) pp. 483ff |

[a2] | M. Hall, "The theory of groups" , Macmillan (1959) pp. Sects. 10.1; 10.5 |

**How to Cite This Entry:**

Supersolvable group.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Supersolvable_group&oldid=43604

This article was adapted from an original article by N.N. Vil'yams (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article