Difference between revisions of "Abelian automorphism group not implies cyclic"

From Groupprops
Jump to: navigation, search
 
 
(One intermediate revision by the same user not shown)
Line 1: Line 1:
#redirect [[abelian automorphism group not implies abelian]]
+
{{group property non-implication|
 +
stronger = group whose automorphism group is abelian|
 +
weaker = cyclic group}}
 +
 
 +
==Statement==
 +
 
 +
It is possible to have a [[group whose automorphism group is abelian]] but such that the group is not a [[cyclic group]].
 +
 
 +
==Related facts==
 +
 
 +
===Opposite facts===
 +
 
 +
* [[Finite abelian and abelian automorphism group implies cyclic]]
 +
 
 +
==Facts used==
 +
 
 +
# [[uses::Locally cyclic implies abelian automorphism group]], [[uses::locally cyclic not implies cyclic]]
 +
# [[uses::Abelian and abelian automorphism group not implies locally cyclic]]
 +
# [[uses::Abelian automorphism group not implies abelian]], [[uses::cyclic implies abelian]]
 +
 
 +
==Proof==
 +
 
 +
We can counterexamples generated from Fact (1), (2), or (3).

Latest revision as of 23:32, 20 June 2013

This article gives the statement and possibly, proof, of a non-implication relation between two group properties. That is, it states that every group satisfying the first group property (i.e., group whose automorphism group is abelian) need not satisfy the second group property (i.e., cyclic group)
View a complete list of group property non-implications | View a complete list of group property implications
Get more facts about group whose automorphism group is abelian|Get more facts about cyclic group

Statement

It is possible to have a group whose automorphism group is abelian but such that the group is not a cyclic group.

Related facts

Opposite facts

Facts used

  1. Locally cyclic implies abelian automorphism group, locally cyclic not implies cyclic
  2. Abelian and abelian automorphism group not implies locally cyclic
  3. Abelian automorphism group not implies abelian, cyclic implies abelian

Proof

We can counterexamples generated from Fact (1), (2), or (3).