Group having no proper subgroup of finite index

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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

Definition

A group having no proper subgroup of finite index is a group satisfying the following equivalent conditions:

  1. It has no proper subgroup of finite index.
  2. It has no proper normal subgroup of finite index.

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

Stronger properties

Weaker properties

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

Opposite properties