Difference between revisions of "Supersolvable group"
From Encyclopedia of Mathematics
m |
m |
||
| Line 6: | Line 6: | ||
$$G=G_1\supseteq G_2\supseteq\dots\supseteq G_{n+1}=E,$$ | $$G=G_1\supseteq G_2\supseteq\dots\supseteq G_{n+1}=E,$$ | ||
| − | in which each quotient group $G_{i-1}/G_i$ is cyclic. Every supersolvable group is a [[ | + | 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). |
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
====References==== | ====References==== | ||
| − | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> K. Doerk, T. Hawkes, "Finite soluble groups" , de Gruyter (1992) pp. 483ff</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> M. Hall, "The theory of groups" , Macmillan (1959) pp. Sects. 10.1; 10.5</TD></TR></table> | + | <table> |
| + | <TR><TD valign="top">[a1]</TD> <TD valign="top"> K. Doerk, T. Hawkes, "Finite soluble groups" , de Gruyter (1992) pp. 483ff</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> M. Hall, "The theory of groups" , Macmillan (1959) pp. Sects. 10.1; 10.5</TD></TR> | ||
| + | </table> | ||
Latest revision as of 07:53, 17 July 2025
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).
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=56024
Supersolvable group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Supersolvable_group&oldid=56024
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