Difference between revisions of "Supersolvable group"
From Encyclopedia of Mathematics
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A group $G$ with a finite series of normal subgroups (cf. [[Subgroup series|Subgroup series]]) | A group $G$ with a finite series of normal subgroups (cf. [[Subgroup series|Subgroup series]]) | ||
| − | $$G=G_1\supseteq G_2\supseteq\ | + | $$G=G_1\supseteq G_2\supseteq\dots\supseteq G_{n+1}=E,$$ |
in which each quotient group $G_{i-1}/G_i$ is cyclic. Every supersolvable group is a [[Polycyclic group|polycyclic group]]. Subgroups and quotient groups of a supersolvable group are also supersolvable, and the commutator subgroup of a supersolvable group is nilpotent. A finite group is supersolvable if and only if all of its maximal subgroups have prime index (Huppert's theorem). | in which each quotient group $G_{i-1}/G_i$ is cyclic. Every supersolvable group is a [[Polycyclic group|polycyclic group]]. Subgroups and quotient groups of a supersolvable group are also supersolvable, and the commutator subgroup of a supersolvable group is nilpotent. A finite group is supersolvable if and only if all of its maximal subgroups have prime index (Huppert's theorem). | ||
Latest revision as of 16:54, 30 December 2018
supersoluble group
A group $G$ with a finite series of normal subgroups (cf. Subgroup series)
$$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=31743
Supersolvable group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Supersolvable_group&oldid=31743
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