Metacyclic group

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

Symbol-free definition

A metacyclic group is a group having a cyclic normal subgroup with a cyclic quotient group.

Definition with symbols

A group $G$ is termed metacyclic if there exists a normal subgroup $N$ of $G$ such that both $N$ and $G/N$ are cyclic.

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
Cyclic group

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
supersolvable group
virtually abelian group		metacyclic implies virtually abelian
polycyclic group
metabelian group
solvable group

Facts

• Classification of metacyclic p-groups