# Supersolvable group

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## 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

VIEW: groups satisfying this property | groups dissatisfying this property
VIEW: Related group property satisfactions | Related group property dissatisfactions

## 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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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