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.

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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.