Are all cyclic groups abelian?
Understand the Problem
The question is asking whether every cyclic group is an abelian group. To answer this, we need to understand the definitions of both cyclic groups and abelian groups and their properties.
Answer
YES, all cyclic groups are Abelian.
The final answer is YES, all cyclic groups are Abelian.
Answer for screen readers
The final answer is YES, all cyclic groups are Abelian.
More Information
Cyclic groups are a fundamental concept in abstract algebra. They are generated by a single element, meaning all other elements of the group can be expressed as powers of this generator element. This inherent structure ensures commutativity, thereby making all cyclic groups Abelian.
Tips
A common mistake is assuming all Abelian groups are cyclic, which is not true. While all cyclic groups are commutative (Abelian), not all commutative groups are cyclic.
Sources
- Are cyclic groups always abelian? - Mathematics Stack Exchange - math.stackexchange.com
- Prove that Every Cyclic Group is an Abelian Group - GeeksforGeeks - geeksforgeeks.org
- Cyclic group - Wikipedia - en.wikipedia.org
