Supersolvable group
From Encyclopedia of Mathematics
The printable version is no longer supported and may have rendering errors. Please update your browser bookmarks and please use the default browser print function instead.
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
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