Finite subnormal subgroup
This article describes a property that arises as the conjunction of a subgroup property: subnormal subgroup with a group property (itself viewed as a subgroup property): finite group
View a complete list of such conjunctions
Definition
A subgroup of a group is termed a finite subnormal subgroup if it is finite as a group and subnormal as a subgroup.
Relation with other properties
Stronger properties
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
|---|---|---|---|---|
| Finite characteristic subgroup | the subgroup is both a finite group and a characteristic subgroup | (characteristic implies subnormal, via normal) | use normal not implies characteristic (finite examples) | |FULL LIST, MORE INFO |
| Finite normal subgroup | the subgroup is both a finite group and a normal subgroup | (normal implies subnormal) | normality is not transitive (finite examples) | |FULL LIST, MORE INFO |
| Subnormal subgroup of finite group | subnormal subgroup where the whole group is a finite group | |FULL LIST, MORE INFO |
Weaker properties
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
|---|---|---|---|---|
| Finitely generated subnormal subgroup | |FULL LIST, MORE INFO | |||
| Subnormal subgroup | |FULL LIST, MORE INFO |