Cyclic Abelian Groups: True for All Cases?

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Discussion Overview

The discussion revolves around the relationship between cyclic and abelian groups in group theory, specifically questioning whether the property of being abelian holds true for all cases of cyclic groups. The scope includes theoretical aspects of group properties.

Discussion Character

  • Debate/contested

Main Points Raised

  • Some participants assert that while cyclic groups are always abelian, the converse is not true, as there exist abelian groups that are not cyclic.
  • One participant seeks clarification on whether the original question intended to imply a relationship rather than equate the two properties.
  • A participant expresses appreciation for the property that cyclic groups are abelian, highlighting its utility in group theory.

Areas of Agreement / Disagreement

Participants generally agree that cyclic groups imply abelian groups, but there is no consensus on the broader implications or definitions, as some participants question the phrasing of the original inquiry.

Contextual Notes

Some statements rely on foundational definitions of group theory, and the discussion does not resolve the nuances of these definitions or their implications.

johnnyboy2005
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is this true for all cases? i know something can be abelian and not cyclic. thanks
 
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It's not true, as you say: some can be abelian but not cyclic. So cyclic implies abelian but not necessarily the other way arround.
 
johnnyboy2005 said:
is this true for all cases? i know something can be abelian and not cyclic. thanks

do you perhaps mean implies instead of =?
 
yes. other wise my question answers itself. so cyclic implies abelian. thanks for the help.
 
you have proved this (elementary in the sense of asked on the first examples sheet of any course in group theory if at all) result...
 
I love this property.

You work with cyclic groups so often that it's so nice to have them all abelian. I love it. :)
 

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