Understanding Cyclic Groups: Why Every Cyclic Group is Always Abelian

Introduction

In the realm of abstract algebra, the concepts of cyclic and abelian groups play a crucial role. A cyclic group is one that is generated by a single element, and an abelian group is one in which the group operation is commutative. It is a common misconception that cyclic groups can be non-abelian, but this is fundamentally incorrect. In this article, we will delve into the true nature of cyclic groups and why they are always abelian.

Definition and Properties of Cyclic Groups

A group is called cyclic if it can be generated by a single element. Formally, a group (G) is cyclic if there exists an element (a in G) such that every element of (G) can be written as (a^n) for some integer (n). This element (a) is called a generator of (G). The group (G) generated by (a) is denoted by (G langle a rangle).

Since cyclic groups are generated by a single element, it follows that any two elements in the group can be expressed in terms of this generator. Let (x a^m) and (y a^n) for some integers (m) and (n). The group operation in a cyclic group is simply exponentiation modulo the order of the group. Therefore, (x cdot y a^m cdot a^n a^{m n}), and by the properties of exponents, (x cdot y a^{m n} a^{n m} a^n cdot a^m y cdot x). This shows that the group operation is commutative, and thus every cyclic group is abelian.

Geometric Example: The Dihedral Group D8

Consider the group of symmetries of a square, often named (D_8), the dihedral group of order 8. This group (D_8) can be described as:

2 axial symmetries with axes joining opposite vertices2 axial symmetries with axes joining the middle of opposite edges4 rotations centered at the center of the square with angles (pi/2), (pi), (3pi/2), and (0) (the latter being the identity neutral element)

While (D_8) is not a cyclic group because no single reflection or rotation can generate all the symmetries, it offers an excellent example of why many cyclic subgroups can be abelian. For instance:

Any axial symmetry (S) generates an abelian subgroup of order 2, ({I, S}), where (S^2 I) and is isomorphic to (mathbb{Z}/2mathbb{Z}).Any non-trivial rotation (R) generates an abelian subgroup of order 4, ({I, R, R^2, R^3}), where (R^4 I) and is isomorphic to (mathbb{Z}/4mathbb{Z}).

Note that even though (D_8) itself is not abelian, these cyclic subgroups are abelian by construction.

Proving the Cyclic Group is Always Abelian

To further solidify our understanding, let's consider a more formal proof that every cyclic group is abelian. We start by defining a group homomorphism from the set of integers (mathbb{Z}) to a cyclic group (G langle a rangle). Let (f: mathbb{Z} to G) be a homomorphism defined by (f(n) a^n) for every integer (n), where (a^{-1} a^{-n}) for (n eq 0). This homomorphism is surjective because every element of (G) can be written as (a^n). Since (mathbb{Z}) is abelian, the homomorphic image of an abelian group is also abelian. Therefore, (G) is isomorphic to a subgroup of (mathbb{Z}), which implies that (G) is abelian.

Additionally, we can use the concept of the multiplicative group of (N)-th roots of unity. If (G) is a finite cyclic group of order (N), then (G) is isomorphic to the multiplicative group of the (N)-th roots of unity. Since these roots are complex numbers and their multiplication is commutative, it follows that the group is abelian.

Conclusion

In conclusion, the claim that there exist cyclic groups that are not abelian is false. Every cyclic group, whether finite or infinite, is always abelian. This is a fundamental result in abstract algebra, and understanding it can provide deeper insights into the properties of groups and their applications in various mathematical and real-world contexts.