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, invariant under isomorphism
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VIEW RELATED: Group property implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions

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

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.