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

This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
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## 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.

## 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 $G = GL(2,3)$, 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.