This article defines a subgroup property: a property that can be evaluated to true/false given a group and a subgroup thereof, invariant under subgroup equivalence. View a complete list of subgroup properties[SHOW MORE]
This article is about a basic definition in group theory. The article text may, however, contain advanced material.
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Definition with symbols
The opposite of the property of being a proper subgroup is the property of being the improper subgroup, viz the whole group.
Relation with other properties
This subgroup property is left-hereditary: any subgroup of a subgroup with this property also has this property. Hence, it is also a transitive subgroup property.
Any subgroup of a proper subgroup is proper. That is, if and is proper in , so is .
This subgroup property is right-hereditary: if a subgroup has the property in a group, it has the property in every bigger group. Hence, it is also a transitive subgroup property.
Any proper subgroup of a subgroup is proper. That is, if and is proper in , is also proper in .