Solvable group generated by finitely many periodic elements
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This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
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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
Contents
Definition
A solvable group generated by finitely many periodic elements is a group that is both a solvable group and a group generated by finitely many periodic elements.
Relation with other properties
Stronger properties
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
|---|---|---|---|---|
| finite abelian group | |FULL LIST, MORE INFO | |||
| finite nilpotent group | |FULL LIST, MORE INFO | |||
| finite solvable group | |FULL LIST, MORE INFO |
Conjunction with other properties
| Conjunction | Other components of conjunction | Explanation |
|---|---|---|
| finite solvable group | could be finite group or periodic group |
Incomparable properties
- periodic solvable group: Clearly, being periodic and solvable does not implies being generated by finitely many periodic elements, because the group may not be finitely generated. For the converse non-implication, see solvable and generated by finitely many periodic elements not implies periodic.
Weaker properties
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
|---|---|---|---|---|
| finitely generated solvable group | solvable and a finitely generated group | any finitely generated torsion-free solvable group, such as the group of integers | |FULL LIST, MORE INFO | |
| solvable group generated by periodic elements | any restricted direct power of infinitely many copies of a nontrivial finite solvable group will do | |FULL LIST, MORE INFO | ||
| solvable group | Solvable group generated by periodic elements|FULL LIST, MORE INFO | |||
| group generated by finitely many periodic elements | |FULL LIST, MORE INFO | |||
| group generated by periodic elements | Solvable group generated by periodic elements|FULL LIST, MORE INFO |
