Abstract algebra is that branch of mathematics which is related to the abstract structure of algebra as compared to the ordinary number systems. The most important of these algebraic structures are Groups, Rings, Fields and Vector spaces, on which the whole abstract algebra depends.

## What is Abstract Algebra

Abstract algebra is the study of the algebraic structure, which starts from the basic and the simplest of all structures, which is groups, and then these groups are extended to rings and then finally to the fields. Some of the concepts and definitions of abstract algebra are as follows:
Group:
Definition 1: A non empty set G with a binary composition " * " is called a group if the following conditions are satisfied:

G.1. a * b belongs to G for all a, b belonging to G. (Closure Law).
G.2. a * (b * c) = (a * b) * c for all a, b, c belonging to G. (Associative Law).
G.3. There exists an element e belonging to G, such that
a * e = e * a = a for all a belonging to G. (Existence of identity, where e is called the identity in G).
G.4. For each a belonging G, there exists an element a’ belonging to G such that a * a’ = a’ * a = e. (Existence of inverse, where a’ is called the inverse of a and is written as a’ = a$^{-1}$).
We write a group G with respect to binary composition * as (G, *).

Definition 2: A group (G, *) is called Abelian or commutative if a * b = b * a for all a, b belonging to G.

Definition 3: If a be an element of a group G, then for any positive integer n, a$^n$ is defined as a$^n$ = a * a * a * …. (n times) and a$^0$ = e. Also a$^{-1} = (a$^{-1})^n$. Ring: Definition 4: A ring is a non empty set R with two binary compositions, denoted by + and * respectively, satisfying the following properties: 1). (R, +) will form an abelian group. 2). a, b belongs to R, for all a, b belonging to R. 3). a * (b * c) = (a * b) * c, for all a, b, c, belonging to R. 4). For all a, b, c belonging to R, a * (b + c) = a * b + a * c, which is known as the left distributive law. (b + c) * a = b * a + c * a, which is known as the right distributive law. If all the above mentioned 4 properties are satisfied by a set of numbers belonging to R, then we say, the set is a ring, which is denoted as [R, +, *]. Zero Divisor: A non zero element a of a commutative ring is called a zero divisor if there exists some non zero element b in R such that a * b = 0. A ring [R, +, *] is called a commutative ring if a * b = b * a for all a, b belonging to R. A ring [R, +, *] is called a field if it’s non zero elements form an abelian group with respect to the composition ‘*’ i.e. if R is a field, then a belonging to R, where a not equal to zero, implies a${-1}$belongs to R. Integral Domain: A commutative ring is called an integral domain if it has no zero divisors in it. The following is the diagrammatic representation of the concepts of abstract algebra: A group G is called finite or infinite according as it contains a finite or infinite number of elements. If a group G contains n elements, we say that the order of G is n and write it as o(G) = n. Based on the above mentioned definitions, some of the examples which come under the concept of abstract algebra are: Examples of Groups: 1). (I, +) is a group, where 0 is the identity and the inverse of a belonging to I is –a belonging to I. 2). Similarly, (R, +), (Q, +) are groups. 3). (N, +) is not a group, as the axioms G.3. and G.4. are not satisfied in N. 4). The set I of integers is not a group under usual multiplication, since the axiom G.4. is not satisfied in I. 5). The set R* of all non- zero real numbers is an abelian group under usual multiplication, where I is the identity and the inverse of a belonging to R* is 1/a. 6). (I, +), (R, +), (Q*, .), (R*, .) are infinite abelian groups. Some of the examples of Rings are: 1). The set E of all even integers is a commutative ring without unit element. 2). The set of all 2 by 2 matrices as shown below: M2(R) =$\begin{Bmatrix}
\begin{pmatrix}
a & b\\
c& d
\end{pmatrix}|a,b,c\ and\ d\ belongs\ to R
\end{Bmatrix}$is a non commutative ring under the addition and multiplication of 2 by 2 matrices. 3). The set of integers with two binary compositions * and o defined by a * b = a + b – 1, aob = a + b – ab for all integers a and b is a commutative ring with unity. The following diagram represents the types of Rings: The following are some examples of integral domain and fields: 1). The ring of integers I is an integral domain because for any tow integers a and b: a . b = 0 implies a = 0 or b = 0. 2). Q and R are fields with respect to the usual addition and multiplication. 3). For any prime p, Jp = {0, 1, 2, …., p - 1} is a field with respect to addition and multiplication modulo p. 4). The set C = {a + bi: a, b belonging to R} of complex numbers is a field under the usual addition and multiplication of complex numbers. ## Abstract Algebra Help Abstract algebra helps to represent problems rather than their result, as this theory adopts an inductive approach of study leading from examples to the general theory. Abstract algebra also helps in analyzing the situations and then looking for their patterns, thus making examples and formulating and then testing conjectures, to prove various theorems. For example, look at the following mapping of a group and its subgroup: This is the group homomorphism from group G to Subgroup H, where N is the kernel of subgroup and aN is one of the coset of N. Abstract algebra helps in finding out the difference between the structures of a ring and ideal, which is helpful in identifying the structures of various chemical bonds, by looking at the following diagram: Here, we can see from the structure itself, that if a and b are two elements of a subring then ab also belongs to the subring, will be a ring. But if a and b belonging to a ring, implies they are commutative, and then it will be considered as an ideal. In mathematics.The concept of prime ideals is used very widely as shown below: These structures of prime ideals help in making the decision regarding the designing of the various computer softwares. The concept of abstract algebra helps in the discovery of various axiomatic systems, as there are various axioms in abstract algebra itself. Abstract algebra also helps in the combinatorics and number theory, and other important areas of mathematics. We use the results of abstract algebra to prove the other important theorems and results which are useful and helpful in solving various problems. For example: Chinese Remainder Theorem: The statement of this theorem is as follows: Let n and m be positive integers, with (n, m) = 1, that is, their GCD should be equal to 1. Then the system of congruences x congruent to a (mod n) and x congruent to b (mod m) has a solution, and that two solutions are congruent to modulo mn. Now, we can use the above theorem of abstract algebra which in turn help us to solve the following problem of finding out the value of the congruence 7x congruent to 3 modulo 5. We will find x such that 5/(7x - 3)…(i) We see that x = 4 is the first positive integer such that (i) is true. Similarly, 4 + 5 = 9, 9 + 5 = 14, etc. all satisfy (i). x = 4 - 5 = -1, - 1 - 5 = -6, - 6 - 5 = -11 etc also satisfy (i). Hence, the solution set of the given congruence is S = {…., -11, -6, -1, 4, 9, 14, 19, ….}. ## Applications of Abstract Algebra There are various applications of abstract algebra present in our real life, not in the filed of mathematics but also in the field of science too, especially, physics and chemistry, in astronomy and in the field of computer sciences where coding and decoding of data is required. Abstract algebra, obviously has a wide application in the filed of mathematics, as the number system which we study form the very basic elementary algebra forms a group under addition. All the numbers which are not equal to zero form a group under multiplication. All the finite groups are basically a set of matrices. Groups of permutation are the building blocks for the theory of determinants and combinatorics. Abstract algebra is used in graph theory too by considering the geometry over some finite fields, in which geometrical objects forms a group of symmetries. For example, consider the following wallpaper structure in R$^2\$:

This pattern can be made very easily by applying the concept of abstract algebra especially lattices in real number system.

The famous Chinese remainder Theorem in number theory is related to the solution of the simultaneous congruence systems, which are widely used in software designing. For example, for the structure of Z4 into Z6, all of the possible substructures and elements can be solved using the concept of abstract algebra as shown below:

This structure can help in debugging the entire software programmed on the hard disk to eliminate the bugs, and thus making the software bug free, in a very easy manner.

## Abstract Algebra Problems

The following are the types of problems which are there in the concept of abstract algebra:

Problem 1: Show that H = {1, -1} is a normal subgroup of the group G = {1, -1, i, -i}
Hint: (A subgroup N of a group G is said to be a normal subgroup of G if gng-1 belongs to N for each g belonging to G and n belonging to N)

Problem 2: Give an example of an infinite group having a finite number of left cosets.
Hint: ((Z, +) has two left cosets Even set of integers and Odd set of integers)

Problem 3: Show that if H is a subgroup of a group G and a, b belonging to G, then
1).    Ha = Hb implies a.b-1 belongs to H
2).    aH = bH if and only if a-1.b belongs to H.
Hint: (Use identity element e and definition of cosets to prove the above theorem)

Problem 4: If R is a ring and a, b belongs to R, then prove that (a + b)2 = a2 + ab + ba + b2.
Hint: ((a + b)2 = (a + b). (a + b) = a (a + b) + b (a + b))

Problem 5: If a, b, c are any 3 elements in a ring R, prove that
1).    a (b - c) = ab – ac,
2).    (a - b) c = ac - bc
Hint: ( a (b - c) =a {b + (-c)} = ab + a (-c) = ab -ac )

## Abstract Algebra Solutions

The following are some of the solutions of abstract algebra questions, which will provide the procedure to adopt while solving any abstract algebra problem:

Question 1: Prove that the set S of all ordered pairs (a, b) of real numbers is a commutative ring under the addition and multiplication compositions defined as (a + b) + (c + d) = (a + c, b + d) and (a, b) (c, d) = (ac, bd).
Solution: We have (a + b) + (c + d) = (a + c, b + d).
It is clear that (S, +) is an abelian group in which (0, 0) is the identity, since (a, b) + (0, 0) = (a + 0, b + 0) = (a, b) and (-a, -b) is the additive inverse of (a, b).
Let x = (a, b), y = (c, d) and z = (e, f) belonging to S.
Then xy = (a, b) (c, d) = (ac, bd) belonging to S. Also xy = yx.
It can be verified that (xy) z = x (yz).
Now, x (y + z) = (a, b) (c + e, d + f) by above definition of addition composition given in the question.
This implies, (a (c + e), b (d + f)) = (ac + ae, bd + bf) = (ac, bd) + (ae, bf).
Therefore, x (y + z) = xy + xz.
Hence, R is a commutative ring.

Question 2: Show that the set I of all integers with binary operation, defined as a.b = a + b + 1, for all a, b belonging to I is an abelian group.
Solution: It is given that a.b = a + b + 1 for all a, be belonging to I.
Let a, b, c be any three integers.
1.    Closure Law: We have a.b = a + b + 1 belongs to I.
2.    Commutative Law: We have a.b = a + b + 1, this implies, b.a = b + a + 1, this implies, a . b = b . a.
3.    Associative Law: We have, (a . b) . c = (a + b + 1) . c, this implies, (a + b + 1) + c + 1 = a + (b + c + 1) + 1, by associative and commutative laws, a . x = a + x + 1, where x = b + c + 1 and b . c = a . x = a . (b . c)
4.    Identity Element: Clearly e = -1 belonging to I, is the identity, since a . e = a + e + 1 = a - 1 + 1 = a for all a belonging to I.
5.    Inverse Law: Consider a. (- a . - 2) = a + (-a - 2) + 1 = -1 = e.
Therefore, inverse of a is - a - 2 for all a belonging to I.
Hence, (I, .) is an abelian group.

Similarly, we can have solutions of many abstract algebra problems using just the definitions of groups, rings or fields, as required by the question.