DefinitionArithmetic functionsParticular casesMetapropertiesRelation with other properties
This property is a pivotal (important) member of its property space. Its variations, opposites, and other properties related to it and defined using it are often studied
Stronger properties
Weaker properties
Property 
Meaning 
Proof of implication 
Proof of strictness (reverse implication failure) 
Intermediate notions


abelian group 
any two elements commute 
cyclic implies abelian 
abelian not implies cyclic 
Epabelian group, Locally cyclic group, Residually cyclic groupFULL LIST, MORE INFO

metacyclic group 
has a cyclic normal subgroup with a cyclic quotient group 
(obvious) 
metacyclic not implies cyclic 
Characteristically metacyclic group, Group with metacyclic derived seriesFULL LIST, MORE INFO

polycyclic group 
has a subnormal series where all the successive quotient groups are cyclic groups 


Characteristically metacyclic group, Characteristically polycyclic group, Finitely generated abelian group, Metacyclic groupFULL LIST, MORE INFO

locally cyclic group 
every finitely generated subgroup is cyclic 


FULL LIST, MORE INFO

group whose automorphism group is abelian 

cyclic implies abelian automorphism group 
abelian automorphism group not implies abelian 
Locally cyclic groupFULL LIST, MORE INFO

group of nilpotency class two 
the inner automorphism group is abelian 
(via abelian) 
(via abelian) 
Group whose automorphism group is abelian, Group whose inner automorphism group is central in automorphism groupFULL LIST, MORE INFO

nilpotent group 

(via abelian) 
(via abelian) 
Abelian group, Epinilpotent group, Group of nilpotency class two, Group whose automorphism group is nilpotentFULL LIST, MORE INFO

finitely generated group 
has a finite generating set 
cyclic means abelian with a generating set of size one 
any finite noncyclic group such as the Klein fourgroup 
Polycyclic groupFULL LIST, MORE INFO

finitely generated abelian group 
finitely generated and abelian 
follows from separate implications for finitely generated and abelian 
Klein fourgroup is a counterexample. 
FULL LIST, MORE INFO

finitely generated nilpotent group 
finitely generated and nilpotent 
(via finitely generated abelian) 
(via finitely generated abelian) 
Finitely generated abelian groupFULL LIST, MORE INFO

supersolvable group 

(via finitely generated abelian) 
(via finitely generated abelian) 
Characteristically metacyclic group, Characteristically polycyclic group, Finitely generated abelian group, Metacyclic groupFULL LIST, MORE INFO

solvable group 



Abelian group, Metabelian group, Metacyclic group, Nilpotent group, Polycyclic groupFULL LIST, MORE INFO

FactsReferences
Textbook references
Book 
Page number 
Chapter and section 
Contextual information 
View


Abstract Algebra by David S. Dummit and Richard M. Foote, 10digit ISBN 0471433349, 13digit ISBN 9780471433347^{More info} 
54 

formal definition 

Groups and representations by Jonathan Lazare Alperin and Rowen B. Bell, ISBN 0387945261^{More info} 
3 

definition introduced in paragraph 

Topics in Algebra by I. N. Herstein^{More info} 
39 
Example 2.4.3 
definition introduced in example 

A Course in the Theory of Groups by Derek J. S. Robinson, ISBN 0387944613^{More info} 
9 



An Introduction to Abstract Algebra by Derek J. S. Robinson, ISBN 3110175444^{More info} 
47 



Finite Group Theory (Cambridge Studies in Advanced Mathematics) by Michael Aschbacher, ISBN 0521786754^{More info} 
2 



Algebra by Serge Lang, ISBN 038795385X^{More info} 
9 



Algebra (Graduate Texts in Mathematics) by Thomas W. Hungerford, ISBN 0387905189^{More info} 
33 

defined as cyclic subgroup 

Algebra by Michael Artin, ISBN 0130047635, 13digit ISBN 9780130047632^{More info} 
46 

Page 46: leading to point (2.7), Page 47, Point (2.9) 

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Definition links