Statements (26)
| Predicate | Object |
|---|---|
| gptkbp:instanceOf |
gptkb:mathematical_concept
Group theory concept |
| gptkbp:application |
Solvability of polynomial equations by radicals
|
| gptkbp:category |
gptkb:algebra
|
| gptkbp:definedIn |
A group with a subnormal series whose factor groups are abelian
|
| gptkbp:example |
Dihedral group D_4
Symmetric group S_3 Symmetric group S_5 |
| gptkbp:field |
gptkb:Mathematics
gptkb:Group_theory gptkb:Abstract_algebra |
| gptkbp:hasWikipediaPage |
https://en.wikipedia.org/wiki/Solvable_group
|
| https://www.w3.org/2000/01/rdf-schema#label |
Solvable group
|
| gptkbp:introduced |
gptkb:Évariste_Galois
|
| gptkbp:notation |
A group G is solvable if there exists a finite sequence of subgroups {1} = G_0 < G_1 < ... < G_n = G such that G_{i+1}/G_i is abelian for all i
|
| gptkbp:property |
A non-abelian simple group is not solvable
Every abelian group is solvable Every quotient of a solvable group is solvable Every subgroup of a solvable group is solvable A finite group is solvable if and only if its composition factors are abelian |
| gptkbp:relatedTo |
gptkb:Galois_theory
gptkb:Nilpotent_group Abelian group Simple group |
| gptkbp:bfsParent |
gptkb:Algebraic_groups
|
| gptkbp:bfsLayer |
7
|