Cyclic Abelian Groups: True for All Cases?

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SUMMARY

Cyclic groups are always abelian, but the reverse is not true; some abelian groups are not cyclic. This fundamental property is established in group theory, where cyclic groups imply abelian structures. The discussion highlights the importance of understanding these definitions, particularly for those studying introductory group theory concepts. Participants confirmed that the relationship between cyclic and abelian groups is a foundational topic in mathematics.

PREREQUISITES
  • Understanding of group theory fundamentals
  • Familiarity with the definitions of cyclic and abelian groups
  • Basic knowledge of mathematical proofs
  • Experience with elementary examples in group theory
NEXT STEPS
  • Study the properties of abelian groups in detail
  • Explore examples of non-cyclic abelian groups
  • Learn about the implications of group homomorphisms
  • Investigate the structure theorem for finitely generated abelian groups
USEFUL FOR

Mathematics students, particularly those studying group theory, educators teaching abstract algebra, and anyone interested in the properties of algebraic structures.

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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