Subgroup of cyclic group

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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]

Definition

Let G be a group and H a subgroup. We use the term subgroup of cyclic group to describe H in G if G is a cyclic group.


Relation with other properties

Weaker properties

Property Meaning Proof of implication Proof of strictness (Reverse implication failure) Intermediate notions
verbal subgroup cyclic implies every subgroup is verbal (any non-cyclic group as a subgroup of itself) |FULL LIST, MORE INFO
fully invariant subgroup (via verbal) (via verbal) |FULL LIST, MORE INFO
characteristic subgroup (via fully invariant) (via fully invariant) |FULL LIST, MORE INFO
normal subgroup (via characteristic) (via characteristic) Characteristic subgroup, Subgroup of abelian group|FULL LIST, MORE INFO
subgroup of abelian group the whole group is abelian |FULL LIST, MORE INFO
cyclic normal subgroup cyclic and a normal subgroup |FULL LIST, MORE INFO
abelian normal subgroup abelian and a normal subgroup Subgroup of abelian group|FULL LIST, MORE INFO
cyclic characteristic subgroup cyclic and a characteristic subgroup |FULL LIST, MORE INFO
abelian characteristic subgroup abelian and a characteristic subgroup |FULL LIST, MORE INFO