Proving the Existence of Generators in Abelian Groups

Understanding the properties of generators in abelian groups is crucial in abstract algebra. Often, students and enthusiasts in mathematics are curious about specific properties, such as the existence of generators for certain groups. This article aims to clarify a common misconception and provide a detailed proof regarding the generators of the direct product of abelian groups.

What is a Generator?

In group theory, a group G is said to have a generator if there exists an element g in G such that every element of G can be expressed as a power of g. Formally, if G is a group and G { e, g, g^2, ldots }, where e is the identity element, then g is a generator of G. However, not every group has a single generator; some can be generated by multiple elements or even require a finite set of generators.

The Misunderstanding: No Single Generator for (Z_2 times Z_2)

It is often claimed or mistakenly believed that the direct product of cyclic groups (Z_2 times Z_2) cannot be generated by a single element. This notion stems from the structure and properties of these groups. However, the absence of a single generator does not mean that the group lacks generators entirely. Let's explore why this is the case.

Direct Product of Cyclic Groups

The direct product of cyclic groups (Z_2 times Z_2) consists of all ordered pairs ((a, b)) where (a, b in Z_2), with the operations defined component-wise. This means that the group has four elements: ((0,0)), ((0,1)), ((1,0)), and ((1,1)), all modulo 2. The group operation is performed by combining the elements component-wise modulo 2.

Non-Product Generators

While it is true that no single element in (Z_2 times Z_2) can generate all four elements, this does not preclude the possibility of generating the group using multiple elements or even a combination of generators. In fact, (Z_2 times Z_2) can be generated by the set of elements ({(1,0), (0,1)}). This shows that generators in abelian groups are not always singletons but can be combinations of multiple elements.

Proof of Generators in (Z_2 times Z_2)

To formally prove that ({(1,0), (0,1)}) generate (Z_2 times Z_2), we need to show that every element of (Z_2 times Z_2) can be written as a combination of these generators. Let’s consider the elements ((0,0)), ((0,1)), ((1,0)), and ((1,1)) and see how they can be generated by ({(1,0), (0,1)}).

Step-by-step Proof

Element (0,0): This is the identity element and can be trivially generated by any element in the group. Element (0,1): This element is already one of the generators, so it is generated by ((0,1)). Element (1,0): This element is also one of the generators, so it is generated by ((1,0)). Element (1,1): This element can be generated by adding the generators ((0,1)) and ((1,0)). Using the group operation, ((1,1) (1,0) (0,1)).

Since all four elements of (Z_2 times Z_2) can be expressed as combinations of ({(1,0), (0,1)}), these elements are indeed generators of (Z_2 times Z_2).

Conclusion

The assertion that no single element can generate (Z_2 times Z_2) does not imply that the group cannot be generated by generators. In fact, (Z_2 times Z_2) can be generated by multiple elements, as demonstrated by the set ({(1,0), (0,1)}). This example highlights the importance of understanding the different types of generators and the properties of abelian groups.

Keywords

Abelian Group, Generator, Proof