Group having no proper subgroup of finite index
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
A group having no proper subgroup of finite index is a group satisfying the following equivalent conditions:
Apart from the trivial group, any group with this property must be infinite.
Equivalence of definitions
The equivalence follows from Poincare's theorem.
Relation with other properties
- Infinite simple group
|Property||Meaning||Proof of implication||Proof of strictness (reverse implication failure)||Intermediate notions|
|group with finitely many homomorphisms to any finite group|
|group in which every subgroup of finite index has finitely many automorphic subgroups|
|group in which every subgroup of finite index is normal|