*94*

# Abelian Groups in Discrete Mathematics

An abelian group is a type of group in which elements always contain commutative. For this, the group law **o** has to contain the following relation:

As compare to the non-abelian group, the abelian group is simpler to analyze. When the group is abelian, many interested groups can be simplified to special cases. For example, the abelian group contains the conjugacy classes, which have singleton sets and are used to contain one element. The Abelian group has many subgroups, which are normal. It contains a set G which is combined with binary operation o. Here, o is used to take G’s two elements and return a G’s element. The relation of this is described as follows:

### Properties of Abelian Groups:

The abelian group has some properties, which is described as follows:

**Associativity:** Suppose set G contains a binary operation o. The operation o is called to be associative in G if it holds the following relation:

**Identity:** Suppose we have an algebraic system (G, o) and set G contains an element e. That element will be called an identifying element of the set if it contains the following relation:

Here, **element e** can be referred to as an identity element of G, and we can also see that it is necessarily unique.

**Inverse:** Suppose there is an algebraic system (G, o), and it contains an identity e. We will also assume that the set G contains an element x and y. The element y will be called an inverse of x if it satisfies the following relation:

Here, **element y** can also be referred to as inverse of x, and we can also see that it is necessarily unique. The inverse of x can also be referred to as x^{-1}.

**Closure:** Suppose there is a set G, which contains elements, x, y. The operation will be called closure in G if the set contains the following relation:

**(x o y) ∈ G**

*⇒***Commutative:** Suppose set G contains a binary operation o. The operation o is called to be commutative in G if it holds the following relation:

We can define the group by using the above four conditions that are an **association, identity, inverse**, and **closure**. The distinction between the non-abelian and the abelian groups is shown by the final condition that is **commutative**.

### Abelian Groups Examples

The cyclic groups are known as the best and simplest example of an abelian group. We can use a single element to generate the cyclic group, and it is isomorphic to Zn, which can be defined as follows:

Zn, set of integers {0, 1, 2, 3,….., n-1}, with group operation of additional module n.

Element of group contains the cycle generated by the group law’s successive application to generators. **For example,** the power of generator g of Z5 can be described as g^{0},g^{1},g^{2},g^{3},g^{4},g^{5}=g^{0},g^{1},g^{2},g^{3},g^{4},… and it can make the elements {g^{0},g^{1},g^{2},g^{3},g^{4}}. Since g^{a} g^{b} = g^{b} g^{a} = g^{a+b}. All these groups are abelian group.

We can say that all abelian groups are not cyclic, but all cyclic groups are abelian. For instance, Z_{2}×Z_{2}, which shows the Klein four groups are not cyclic but abelian. In contrast with the above law, the abelian group will not form by the invertible matrices group with matrix multiplication group law. It is because for the matrix M and N, MN = NM is not true. If n >=3 in the symbolic group S_{n,} this group will also be non-abelian.

The example of an abelian group can also be described by the rings with their additive operations. Units of rings can use their multiplicative operation and form an abelian group. For example, the additive abelian group can be formed by the real number, and the multiplicative abelian group can be formed by the nonzero real numbers, which is denoted by R^{*}

### Properties of Abelian Groups

The abelian group is used to form many group properties in special cases. For example, we have already mentioned that conjugacy classes are singleton sets. Similarly,

- The group’s center is the same as itself. There is another case that converse is also true, which means the group will be abelian if the group’s center is the same as the group itself.
- Two elements of commutator (g
^{-1}h^{-1}gh) show the identity in an abelian group. - Abelian group has the derived subgroup, which is trivial.

A variety of algebras are also formed by the abelian group. That means:

- Abelian group has subgroups that are also abelian.
- Abelian group has the quotient groups, which are also abelian.
- The two abelian group has the direct product which is also abelian.

Lastly, we will describe the group law in which the set of a homomorphism form another abelian group from an abelian group to another. The syntax of group law is described as follows:

### Abelian Groups Classification

On the basis of order, an abelian group will be classified. The Order of a group will be calculated by the number of elements in a group. A group G will be known as the finite group if the order of a group is finite. Now we will describe **Kronecker’s decomposition theorem**. According to this theorem, we can write the abelian group of order n in the following format:

_{k1}⊕Z

_{k2}⊕…⊕Z

_{km},

Where **k _{i}** is used to show the powers of primes and k

_{i}is multiplied to n. This type of representation is unique.

**For Example:**

If we want to write order 15 in the abelian group, it will be written as Z_{3}⊕Z_{5}. It is used to implying that in order 15, all abelian groups are isomorphic. We will also show a explicit example that is {0,5,10}⊕{0,3,6,9,12}. We also have two more additional special cases, which is described as follows:

- If a group contains an order p, this group will be isomorphic to Zp and necessarily abelian. It is also cyclic.
- If a group contains an order p
^{2}, this group will necessarily be abelian, and it is also isomorphic to either Z_{p}_{2}or Z×Z_{p}._{p}

According to the above Kronecker’s decomposition theorem, it is described that the number of non-isomorphic abelian groups of order (n) has the following relation: n=∏_{i}p_{i}^{ei} and

_{i}P(e

_{i}),

Here,

**a(n)** is used to describe the number of partitions of n, or we can say that **a(n)** is the product of a number of partitions of each exponent in the prime factorization of n.