Locally cyclic implies abelian automorphism group: Difference between revisions

Latest revision as of 18:57, 20 June 2013

This article gives the statement and possibly, proof, of an implication relation between two group properties. That is, it states that every group satisfying the first group property (i.e., locally cyclic group) must also satisfy the second group property (i.e., group whose automorphism group is abelian)
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Statement

Any locally cyclic group is a group whose automorphism group is abelian, i.e., it has an abelian automorphism group.

Related facts

Proof

PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]

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 {{group property implication|
 stronger = locally cyclic group|
-weaker = aut-abelian group}}
+weaker = group whose automorphism group is abelian}}
 ==Statement==
-Any [[locally cyclic group]] is an [[aut-abelian group]], i.e., it has an [[abelian group|abelian]] [[automorphism group]].
+Any [[locally cyclic group]] is a [[group whose automorphism group is abelian]], i.e., it has an [[abelian group|abelian]] [[automorphism group]].
 ==Related facts==
-* [[Cyclic implies aut-abelian]]
+* [[Cyclic implies abelian automorphism group]]
-* [[Abelian and aut-abelian not implies locally cyclic]]
+* [[Abelian and abelian automorphism group not implies locally cyclic]]
-* [[Aut-abelian not implies abelian]]
+* [[Abelian automorphism group not implies abelian]]
-* [[Finite abelian and aut-abelian implies cyclic]]
+* [[Finite abelian and abelian automorphism group implies cyclic]]
 ==Proof==
 {{fillin}}