Cyclic Abelian Groups: True for All Cases?

  • Thread starter johnnyboy2005
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In summary, it is not true that all abelian groups are cyclic. However, cyclic groups are always abelian. This result is often considered elementary in group theory and is commonly seen in introductory examples. Many people appreciate the fact that all cyclic groups are abelian.
  • #1
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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  • #2
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.
 
  • #3
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 =?
 
  • #4
yes. other wise my question answers itself. so cyclic implies abelian. thanks for the help.
 
  • #5
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...
 
  • #6
I love this property.

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

1. What is a cyclic abelian group?

A cyclic abelian group is a group in which every element can be generated by repeatedly applying a single element. In other words, every element in a cyclic abelian group can be expressed as a power of a single element.

2. What are some examples of cyclic abelian groups?

Some examples of cyclic abelian groups include the group of integers under addition, the group of real numbers under addition, and the group of complex numbers under multiplication.

3. Are all cyclic groups also abelian?

Yes, all cyclic groups are also abelian. This is because in a cyclic group, the group operation is commutative, meaning the order in which elements are multiplied does not affect the result.

4. What is the order of a cyclic abelian group?

The order of a cyclic abelian group is the number of elements in the group. This is always finite for a cyclic abelian group, as a cyclic group can be generated by a finite set of elements.

5. Is it true that all cyclic abelian groups have the same structure?

Yes, it is true that all cyclic abelian groups have the same structure. This is because any two cyclic abelian groups of the same order are isomorphic, meaning they have the same underlying structure even if the elements are labeled differently.

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