Can Every Cyclic Group Be Represented as a Homomorphic Image of Z?

A cyclic group is defined as a group that is generated by a single element and its inverses. One of the definitive characteristics of a cyclic group is that it can be represented as a homomorphic image of the integers, denoted by Z.

Proof and Representation of a Cyclic Group

If x is the generator of the group, the function f: Z → x given by fn x^n serves to demonstrate the equivalence of the group elements and their representation as powers of the generator. This is not just evidence but solid proof of the cyclic group's homomorphic image with respect to Z.

Infinite Cyclic Groups

A finite cyclic group can be uniquely represented by its generator and the exponents of the group elements. However, for an infinite cyclic group G generated by an element a, each element in G can be represented as a^n for some integer n. This representation is unique and makes the group operation indistinguishable from addition within the integers.

Multiplying two elements a^n and a^m in the group G under the group operation results in a^na^m a^{nm}. The essential effect is observed in the exponents, which highlights the group operation’s equivalence to integer addition. This equivalence underscores the fundamental nature of the group as a homomorphic image of Z.

Unique Representation and Intuition

It is important to note that there is only one infinite cyclic group, regardless of how it is represented. Representations such as those generated by different generators or with varying exponents ultimately describe the same group. This is akin to the infinite circle expanding as more nodes are added, virtually becoming an infinite line.

Both Z and the cyclic group of infinite order can be represented as the set of integers under any operation that satisfies the group axioms. For example, when Z is considered with the operation of addition, it serves as a cyclic group with a generator 1 and no relations. This highlights the underlying equivalence between these two representations.

Cayley Graph Visualization

The Cayley graph of the cyclic group of order 8 can be illustrated with 0 as the identity and 1 as the generator. This graph visually represents how the group elements are connected through the group operation. As the order of the group increases, the graph (originally a circle) becomes larger. In the limit as the order approaches infinity, the circle transforms into a line, effectively representing the infinite cyclic group.

However, it is crucial to recognize that the concept of an infinite group introduces some peculiarities that set it apart from finite groups. While the extension from finite to infinite groups may aid in building intuition, discussing infinite groups involves dealing with unique behaviors and properties that don't apply to finite counterparts.

In summary, every cyclic group can indeed be represented as a homomorphic image of Z, and this representation helps us understand the fundamental nature of cyclic groups, both finite and infinite. The concept of the infinite cyclic group can be visualized as the set of integers under addition, embodying the core structure of cyclic groups in a unique and profound way.

Keywords: cyclic group, homomorphic image, infinite cyclic group