The Group Properties Wiki (pre-alpha)

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From Groupprops

This article describes a property that arises as the conjunction of a subgroup property: normal subgroup with a group property (itself viewed as a subgroup property): cyclic group
View a complete list of such conjunctions

Contents 1 Definition 1.1 Symbol-free definition 2 Relation with other properties 2.1 Stronger properties 2.2 Weaker properties 3 Facts

Definition

Symbol-free definition

A subgroup of a group is termed a cyclic normal subgroup if it is cyclic as a group and normal as a subgroup.

Relation with other properties

Stronger properties

• Cyclic characteristic subgroup

Weaker properties

• Hereditarily normal subgroup: For full proof, refer: Cyclic normal implies hereditarily normal
• Abelian normal subgroup
• Nilpotent normal subgroup
• Solvable normal subgroup

Facts

• Supersolvable implies every nontrivial normal subgroup contains a cyclic normal subgroup