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
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abelian group |
any two elements commute |
cyclic implies abelian |
abelian not implies cyclic |
Epabelian group, Locally cyclic group, Residually cyclic group|FULL 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 series|FULL LIST, MORE INFO
|
polycyclic group |
has a subnormal series where all the successive quotient groups are cyclic groups |
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Characteristically metacyclic group, Characteristically polycyclic group, Finitely generated abelian group, Metacyclic group|FULL LIST, MORE INFO
|
locally cyclic group |
every finitely generated subgroup is cyclic |
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|FULL LIST, MORE INFO
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group whose automorphism group is abelian |
|
cyclic implies abelian automorphism group |
abelian automorphism group not implies abelian |
Locally cyclic group|FULL 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 group|FULL LIST, MORE INFO
|
nilpotent group |
|
(via abelian) |
(via abelian) |
Epinilpotent group, Group of nilpotency class two, Group whose automorphism group is nilpotent|FULL LIST, MORE INFO
|
finitely generated group |
has a finite generating set |
cyclic means abelian with a generating set of size one |
any finite non-cyclic group such as the Klein four-group |
Polycyclic group|FULL LIST, MORE INFO
|
finitely generated abelian group |
finitely generated and abelian |
follows from separate implications for finitely generated and abelian |
Klein four-group 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 group|FULL LIST, MORE INFO
|
supersolvable group |
|
(via finitely generated abelian) |
(via finitely generated abelian) |
Characteristically metacyclic group, Characteristically polycyclic group, Finitely generated abelian group, Metacyclic group|FULL LIST, MORE INFO
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solvable group |
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Metabelian group, Metacyclic group, Polycyclic group|FULL LIST, MORE INFO
|
FactsReferences
Textbook references
Book |
Page number |
Chapter and section |
Contextual information |
View
|
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Abstract Algebra by David S. Dummit and Richard M. Foote, 10-digit ISBN 0471433349, 13-digit ISBN 978-0471433347More info |
54 |
|
formal definition |
|
Groups and representations by Jonathan Lazare Alperin and Rowen B. Bell, ISBN 0387945261More info |
3 |
|
definition introduced in paragraph |
|
Topics in Algebra by I. N. HersteinMore info |
39 |
Example 2.4.3 |
definition introduced in example |
|
A Course in the Theory of Groups by Derek J. S. Robinson, ISBN 0387944613More info |
9 |
|
|
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An Introduction to Abstract Algebra by Derek J. S. Robinson, ISBN 3110175444More info |
47 |
|
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Finite Group Theory (Cambridge Studies in Advanced Mathematics) by Michael Aschbacher, ISBN 0521786754More info |
2 |
|
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Algebra by Serge Lang, ISBN 038795385XMore info |
9 |
|
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Algebra (Graduate Texts in Mathematics) by Thomas W. Hungerford, ISBN 0387905189More info |
33 |
|
defined as cyclic subgroup |
|
Algebra by Michael Artin, ISBN 0130047635, 13-digit ISBN 978-0130047632More info |
46 |
|
Page 46: leading to point (2.7), Page 47, Point (2.9) |
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External links
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Definition links