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| {{unreferenced|date=June 2013}}
| | Jayson Berryhill is how I'm called and my spouse doesn't like it at all. To play lacross is 1 of the things she loves most. Mississippi is exactly where his house is. She functions as a travel agent but quickly she'll be on her personal.<br><br>my web-site: real psychics ([http://www.aseandate.com/index.php?m=member_profile&p=profile&id=13352970 please click the next internet page]) |
| In [[group theory]], a '''metacyclic group''' is an [[group extension|extension]] of a [[cyclic group]] by a cyclic group. That is, it is a group ''G'' for which there is a [[short exact sequence]]
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| :<math>1 \rightarrow K \rightarrow G \rightarrow H \rightarrow 1,\,</math>
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| where ''H'' and ''K'' are cyclic. Equivalently, a metacyclic group is a group ''G'' having a cyclic [[normal subgroup]] ''N'', such that the [[quotient group|quotient]] ''G''/''N'' is also cyclic.
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| ==Properties==
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| Metacyclic groups are both [[supersolvable group|supersolvable]] and [[metabelian]].
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| ==Examples==
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| * Any [[cyclic group]] is metacyclic.
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| * The [[direct product of groups|direct product]] or [[semidirect product]] of two cyclic groups is metacyclic. These include the [[dihedral group]]s, the [[quasidihedral group]]s, and the [[dicyclic group]]s.
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| * Every [[finite group]] of [[Square-free integer|squarefree]] order is metacyclic.
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| * More generally every [[Z-group#Groups whose Sylow subgroups are cyclic|Z-group]] is metacyclic. A Z-group is a group whose Sylow subgroups are cyclic.
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| ==References==
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| *{{springer|id=M/m063550|title=Metacyclic group|author=A. L. Shmel'kin}}
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| [[Category:Group theory]]
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| [[Category:Properties of groups]]
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| [[Category:Solvable groups]]
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| {{Abstract-algebra-stub}}
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Latest revision as of 13:13, 5 May 2014
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