Are Cyclic Groups Abelian?

Yes, cyclic groups are always abelian. This fundamental property stems directly from their definition.

Understanding Cyclic and Abelian Groups

To fully grasp why cyclic groups are abelian, it's essential to understand what each term means:

• Cyclic Group: A group is called cyclic if it can be generated by a single element. This means every element in the group can be expressed as an integer power of that one generator. For instance, if 'g' is the generator, any element 'x' in the group can be written as x = g^k for some integer k.

  • Learn more about Cyclic Groups

• Abelian Group (Commutative Group): An abelian group is a group in which the result of applying the group operation to two elements does not depend on the order in which they are written. In other words, for any two elements 'a' and 'b' in the group, a * b = b * a. This property is known as commutativity.

  • Learn more about Abelian Groups

The Proof: Why Cyclicity Implies Abellaness

The proof that cyclic groups are abelian is straightforward and relies on the properties of exponents.

Let G be a cyclic group generated by a single element g. This means that every element in G can be written in the form g^k for some integer k (where k can be positive, negative, or zero).

Consider any two arbitrary elements from the group, let's call them a and b. Since G is cyclic and generated by g, we can express a and b as powers of g:

• a = g^m (for some integer m)
• b = g^n (for some integer n)

Now, let's look at their product in both orders:

1. *Product `a b:** a b = g^m g^n`

   Using the standard rules of exponents (which hold in any group context when powers of the same element are involved), we have:
   g^m * g^n = g^(m+n)

2. *Product `b a:** b a = g^n g^m`

   Similarly, using the rules of exponents:
   g^n * g^m = g^(n+m)

Since m and n are integers, integer addition is commutative, meaning m+n = n+m.

Therefore, it follows that:
g^(m+n) = g^(n+m)

Which implies:
a * b = b * a

This demonstrates that any two elements a and b in a cyclic group commute. Because this holds true for all elements in the group, a cyclic group by definition satisfies the condition for being an abelian group.

Illustrative Commutation

This property can be simply visualized:

Examples of Cyclic Abelian Groups

Many common groups are cyclic and thus abelian:

• The Integers under Addition (Z): This group is cyclic, generated by 1 (or -1). For example, 5 can be written as 1+1+1+1+1 (or 1^5 in multiplicative notation, where the operation is addition), and -3 as -1-1-1. All integers commute under addition (e.g., 5 + 3 = 8 and 3 + 5 = 8).

• Integers Modulo n under Addition (Z_n): For any positive integer n, the group Z_n (e.g., Z_5 = {0, 1, 2, 3, 4} under addition modulo 5) is cyclic. It is generated by 1 (or any element relatively prime to n). For example, in Z_5, 2 = 1+1, 3 = 1+1+1, etc. All elements in Z_n commute under addition modulo n.

• The Group of Roots of Unity: The n-th roots of unity under multiplication form a cyclic group, which is also abelian.

All these examples reinforce the universal truth that if a group is cyclic, it must necessarily be abelian.