Supersolvable group

Definition

Symbol-free definition

A group is said to be supersolvable if it has a normal series (wherein all the members are normal in the whole group) of finite length, starting from the trivial group and ending at the whole group, such that all the successive quotients are cyclic.

Definition with symbols

A group $G$ is said to be supersolvable if there exists a normal series:

$1 = H_0 \le H_1 \le H_2 \le \ldots \le H_n = G$

where each $H_i \triangleleft G$ and further, each $H_{i+1}/H_i$ is cyclic.

Examples

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Metaproperties

Metaproperty name	Satisfied?	Proof	Statement with symbols
pseudovarietal group property	Yes	supersolvability is pseudovarietal	Supersolvability is closed under taking subgroups, quotients, and finite direct products (see below).
subgroup-closed group property	Yes	supersolvability is subgroup-closed	If $G$ is supersolvable, and $H \le G$ is a subgroup, $H$ is supersolvable.
quotient-closed group property	Yes	supersolvability is quotient-closed	If $G$ is supersolvable, and $H$ is a normal subgroup of $G$, the quotient group $G/H$ is supersolvable.
finite direct product-closed group property	Yes	supersolvability is finite direct product-closed	If $G_1,G_2,\dots,G_n$ are all supersolvable, the external direct product $G_1 \times G_2 \times \dots \times G_n$ is also supersolvable.

This article is about a standard (though not very rudimentary) definition in group theory. The article text may, however, contain more than just the basic definition
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This article defines a group property that is pivotal (i.e., important) among existing group properties
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VIEW RELATED: Group property implications | Group property non-implications | Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions |Group property dissatisfactions
RANDOM GROUP PROPERTY: Simple group: A nontrivial group having no proper nontrivial normal subgroup.

This is a variation of solvability|Find other variations of solvability |

The version of this for finite groups is at: finite supersolvable group

Relation with other properties

Conjunction with other properties

Conjunction	Other component of conjunction
finite supersolvable group	finite group

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
finite abelian group				Finitely generated abelian group|FULL LIST, MORE INFO
finitely generated abelian group				Finitely generated nilpotent group|FULL LIST, MORE INFO
finite nilpotent group				Finitely generated nilpotent group|FULL LIST, MORE INFO
finitely generated nilpotent group				|FULL LIST, MORE INFO
finite supersolvable group				|FULL LIST, MORE INFO

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
polycyclic group	has a subnormal series with cyclic quotients			|FULL LIST, MORE INFO
solvable group				Polycyclic group|FULL LIST, MORE INFO
finitely generated solvable group				Polycyclic group|FULL LIST, MORE INFO
group with nilpotent derived subgroup		Supersolvable implies nilpotent commutator subgroup

Study of the notion

Mathematical subject classification

Under the Mathematical subject classification, the study of this notion comes under the class: 20F16