Group in which every nontrivial normal subgroup contains a cyclic normal subgroup

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This article defines a group property: a property that can be evaluated to true/false for any given group
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RANDOM GROUP PROPERTY: SQ-universal group: A group for which every finitely generated group can be realized as a subquotient.

Definition

A group in which every nontrivial normal subgroup contains a cyclic normal subgroup is a group with the property that any nontrivial normal subgroup of the group, contains a cyclic normal subgroup.

Relation with other properties

Stronger properties

Abelian group
Nilpotent group: For full proof, refer: Nilpotent implies every nontrivial normal subgroup contains a cyclic normal subgroup
Group whose center is normality-large: For full proof, refer: Center is normality-large implies every nontrivial normal subgroup contains a cyclic normal subgroup
Supersolvable group: For full proof, refer: Supersolvable implies every nontrivial normal subgroup contains a cyclic normal subgroup

Metaproperties

Quotients

This group property is not quotient-closed, viz., we could have a group with the property and a quotient group of that group that does not have the property

It can happen that a group $G$ satisfies the property that every nontrivial normal subgroup contains a cyclic normal subgroup, but the quotient of $G$ doesn't satisfy that property. For instance, if we take $G$ to be a quasisimple group that is not simple, then its center is normality-large, so $G$ satisfies the property, but $G / Z (G)$ , being simple, doesn't. In a similar vein, if we take $G = G L (2,3)$ , then the center of $G$ is normality-large, and $G / Z (G)$ , which is isomorphic to the symmetric group on four elements, does not satisfy the property. (in fact, it has no cyclic normal subgroups).

Facts

If $G$ has a subgroup $H$ that is normality-large, transitively normal, and satisfies this property (namely, every nontrivial normal subgroup of $H$ contains a cyclic normal subgroup), then $G$ also satisfies the property. This generalizes the observation that if the center is normality-large, then the group satisfies the property.