Metacyclic group
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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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Contents
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 is termed metacyclic if there exists a normal subgroup
of
such that both
and
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 |