Group in which every nontrivial normal subgroup has 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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Definition
A group in which every nontrivial normal subgroup has finite index is a group with the property that every nontrivial normal subgroup (i.e., every normal subgroup other than the trivial subgroup) is a subgroup of finite index: in other words, its index in the whole group is finite.
Relation with other properties
Stronger properties
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
|---|---|---|---|---|
| finite group | |FULL LIST, MORE INFO | |||
| simple group | |FULL LIST, MORE INFO |
Weaker properties
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
|---|---|---|---|---|
| group in which every normal subgroup is finite or has finite index | |FULL LIST, MORE INFO | |||
| residually finite group unless the group is simple |