Supersolvable group
From Groupprops
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 is said to be supersolvable if there exists a normal series:
where each and further, each
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 ![]() ![]() ![]() |
quotient-closed group property | Yes | supersolvability is quotient-closed | If ![]() ![]() ![]() ![]() |
finite direct product-closed group property | Yes | supersolvability is finite direct product-closed | If ![]() ![]() |
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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View a complete list of semi-basic definitions on this wiki
This article defines a group property that is pivotal (i.e., important) among existing group properties
View a list of pivotal group properties | View a complete list of group properties [SHOW MORE]
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