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Difference between revisions of "Supersolvable group"

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''supersoluble group''
 
''supersoluble group''
  
A group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091260/s0912601.png" /> with a finite series of normal subgroups (cf. [[Subgroup series|Subgroup series]])
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A group $G$ with a finite series of normal subgroups (cf. [[Subgroup series|Subgroup series]])
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091260/s0912602.png" /></td> </tr></table>
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$$G=G_1\supseteq G_2\supseteq\dots\supseteq G_{n+1}=E,$$
  
in which each quotient group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091260/s0912603.png" /> 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).
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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=12272
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