Subgroup of cyclic group

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]

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VIEW FACTS THAT:use, or start with, a subgroup satisfying the property|prove, or show that, a subgroup satisfies the property

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|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			|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