Give an example that every abelian group is not cyclic.

To show that **not every abelian group is cyclic**, let's use a counterexample.

### Example: The Group \( \mathbb{Z}_2 \times \mathbb{Z}_2 \)
Consider the group \( G = \mathbb{Z}_2 \times \mathbb{Z}_2 \), which is the direct product of two copies of the cyclic group \( \mathbb{Z}_2 \). 

1. **Elements of the group**:  
   The elements of \( G \) are pairs \( (a, b) \), where \( a \) and \( b \) are elements of \( \mathbb{Z}_2 \). Since \( \mathbb{Z}_2 = \{0, 1\} \), the group \( G \) has the following elements:
   \[
   G = \{(0, 0), (0, 1), (1, 0), (1, 1)\}
   \]
   This group has 4 elements.

2. **Abelian property**:  
   \( G \) is **abelian** because the group operation (component-wise addition mod 2) is commutative:
   \[
   (a_1, b_1) + (a_2, b_2) = (a_2, b_2) + (a_1, b_1) \quad \text{for all} \quad (a_1, b_1), (a_2, b_2) \in G
   \]
   
3. **Cyclic property check**:  
   To be **cyclic**, there must exist an element \( g \in G \) such that all elements of \( G \) can be written as powers of \( g \) (or, in additive notation, multiples of \( g \)).

   - Let’s check each element to see if it generates the entire group:
     - If \( g = (0, 0) \), the only element you get is \( (0, 0) \), which does not generate the whole group.
     - If \( g = (1, 0) \), the possible multiples of \( g \) are:
       \[
       (1, 0), (1, 0) + (1, 0) = (0, 0)
       \]
       So, it only generates \( \{(0, 0), (1, 0)\} \), which is not the whole group.
     - Similarly, if \( g = (0, 1) \), it only generates \( \{(0, 0), (0, 1)\} \), not the whole group.
     - Finally, if \( g = (1, 1) \), the multiples are:
       \[
       (1, 1), (1, 1) + (1, 1) = (0, 0)
       \]
       This only generates \( \{(0, 0), (1, 1)\} \), which is also not the whole group.
   
4. **Conclusion**:  
   Since no single element generates all four elements of \( G \), this group is **not cyclic**. Therefore, \( \mathbb{Z}_2 \times \mathbb{Z}_2 \) is an abelian group that is not cyclic.

This provides an example showing that **not all abelian groups are cyclic**.

---

Would you like more details or have any questions about this?

### 5 Related Questions:
1. What is the difference between cyclic and abelian groups?
2. Can you give an example of a cyclic abelian group?
3. What is the structure of direct product groups like \( \mathbb{Z}_2 \times \mathbb{Z}_2 \)?
4. How do you determine if a group is cyclic?
5. Are all finite abelian groups cyclic?

### Tip:
For finite abelian groups, the **Fundamental Theorem of Finite Abelian Groups** states that every finite abelian group is a direct product of cyclic groups, but not necessarily a single cyclic group.

Give an example that every abelian group is not cyclic.

Understand why not every abelian group is cyclic through the example of the group Z2 x Z2. This detailed explanation covers group theory concepts, checks for cyclic properties, and provides insights into the structure of abelian groups. Suitable for students studying abstract algebra at the undergraduate level.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation