Are All Subgroups of Abelian Groups Cyclic?

It is a common question in group theory whether every subgroup of an abelian group is cyclic. This article aims to explore this question and provide a detailed explanation to clarify the concepts involved.

Introduction

Let's start with the basics. An abelian group is a group in which the binary operation is commutative. A subgroup of an abelian group is a subset that is itself a group under the same operation. The question often asked is whether every such subgroup must be cyclic. In this article, we will provide a clear answer and illustrate it with examples and proofs.

Key Concepts

Abelian Groups

An abelian group is a structure ((G, )) where (G) is a set and ( ) is a binary operation on (G) that satisfies the following properties:

Associativity: ((a b) c a (b c)) for all (a, b, c in G) Commutativity: (a b b a) for all (a, b in G) Existence of identity: (exists e in G) such that (a e a) for all (a in G) Existence of inverses: (forall a in G exists a^{-1} in G) such that (a a^{-1} e)

Cyclic Subgroups

A subgroup (H) of an abelian group (G) is cyclic if there exists an element (a in G) such that every element of (H) can be written as (a^n) for some integer (n). This element (a) is called a generator of the subgroup (H).

The Question Revisited

The answer to the question is No. Not every subgroup of an abelian group is cyclic. While it is true that every subgroup of a cyclic group is cyclic, this is not a universal property of all abelian groups.

Example: (mathbb{Z} times mathbb{Z})

Consider the abelian group (mathbb{Z} times mathbb{Z}), which is the direct product of two copies of the integers. This group is abelian and can be thought of as the set of all ordered pairs of integers under component-wise addition.

One of its subgroups is the set (H { (0, 0), (1, 0), (0, 1), (1, 1) }). This subgroup is isomorphic to (mathbb{Z}/2mathbb{Z} times mathbb{Z}/2mathbb{Z}), a non-cyclic group. There is no single element in (H) that can generate all other elements of (H), thus demonstrating that (H) is not cyclic.

Another Example: (mathbb{Z}_2 times mathbb{Z}_2 times mathbb{Z}_2)

Another example where a subgroup is not cyclic is (mathbb{Z}_2 times mathbb{Z}_2 times mathbb{Z}_2). A subgroup of this group is (mathbb{Z}_2 times mathbb{Z}_2), which is itself not cyclic. This can be confirmed by the fact that there is no single element that can generate all others.

Non-Cyclic Abelian Groups

Furthermore, there are abelian groups that are not cyclic, and thus their subgroups can also be non-cyclic. For instance, consider the finite abelian group (mathbb{Z}_5). Although (mathbb{Z}_5) is cyclic, it can have subgroups that are not cyclic. These examples show that the property of cyclic subgroups is not universal in abelian groups.

Conclusion

In summary, it is possible for a subgroup of an abelian group to be non-cyclic. While cyclic subgroups are a property of cyclic groups, they are not a universal property of all abelian groups. Understanding the structure of subgroups in abelian groups is crucial for a deep dive into group theory.

Keywords: abelian group, cyclic subgroup, non-cyclic subgroup