Supersolvable group

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> @@@
RANDOM GROUP PROPERTY: <random>Simple group: A nontrivial group having no proper nontrivial normal subgroup.@@@Noetherian group: A group where every subgroup is finitely generated.@@@Complete group: A group for which the natural homomorphism to its automorphism group is an isomorphism, i.e., a centerless group for which every automorphism is inner.@@@SQ-universal group: A group for which every finitely generated group can be realized as a subquotient.@@@T-group: A group where any subnormal subgroup is normal, i.e., where normality is transitive. In general, normality is not transitive.@@@Locally finite group: A group in which every finitely generated subgroup is finite.@@@Group satisfying normalizer condition: A group with no proper self-normalizing subgroup.</random></random>

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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:

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

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

Relation with other properties

Stronger properties

• Nilpotent group

Weaker properties

• Polycyclic group
• Solvable group
• Monomial-representation group
• Nilpotent-commutator group

Facts

Derived subgroup is nilpotent

It turns out that for any supersolvable group, the commutator subgroup is nilpotent.

Normal Abelian subgroup properly containing the center

In a supersolvable group, there is a normal Abelian group property containing the center. This fact turns out to be crucially important for proving that every supersolvable group is a monomial-representation group.

Every representation is monomial

It turns out that every representation of a supersolvable group is monomial. In other words, every irreducible representation of a supersolvable group can be induced from a one-dimensional representation of some subgroup.

Metaproperties

Subgroups

This group property is subgroup-closed, viz., any subgroup of a group satisfying the property also satisfies the property
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Any subgroup of a supersolvable group is supersolvable. The normal series for the subgroup can be obtained simply by intersecting the normal series of the group, with the subgroup.

Quotients

This group property is quotient-closed, viz., any quotient of a group satisfying the property also has the property
View a complete list of quotient-closed group properties

Any quotient of a supersolvable group is supersolvable. The normal series for the quotient is obtained by taking the image of the normal series for the original group via the quotient map.

Any direct product of supersolvable groups is supersolvable. In fact, more generally, any central product of supersolvable groups is supersolvable.