Group having no proper subgroup of finite index

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.

Stronger properties

 * Infinite simple group

Opposite properties

 * Residually finite group
 * Profinite group
 * Group in which every nontrivial normal subgroup has finite index