What is a Group in Abstract Algebra?

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In the 1800s, mathematicians knew how to solve equations like these starting from the linear equation.
However, what about equations of higher degree? Degrees 5,6,7 and beyond?
A young teenager at the time in France, Ä’variste Galois answered this question. .


And to do so he used a tool that he called a “group”.
Around this time, Carl Friedrich Gauss was making sensational discoveries of his own. He showed a new technique called modular arithmetic which helped him solve many problems in number theory.



As it turned out, Modular arithmetic shared many similarities to the groups used by Galois.The 1800s also saw a revolution in geometry. For more than 2000 years, Euclid dominated the scene with his book – “The Elements” but mathematicians began to realize there are other geometries beyond the one devised by the ancient Greeks. It didn’t take long before groups were found to be a useful tool in studying these new geometries.



It soon became clear that groups were a powerful tool that could be used in many different areas of mathematics. So it made sense to “abstract” out the common features of this tool used by Galois, Gauss and others into a general tool, and to learn everything about it.

This leads us to the question -what is a group?
It’s an abstract idea, but what do you expect from abstract algebra? Because groups are abstract, they generalize a lot of different things from arithmetic, algebra, geometry and more. This generality is what makes groups so powerful. The mathematics we learn in abstract algebra can be applied to many other areas.

In arithmetic, we add, subtract, multiply and divide different kind of numbers.
+   ̶   × ÷

 They can be integers, fractions (Rational numbers), real numbers or even complex numbers.
š¯›§: Integers
Q: Fractions
R: Real numbers
C: Complex numbers

However, notice that we can reduce two of the operations.
7-4=7+(-4)
9÷5=9×(1/5)

So in arithmetic, we only really have two operations. Also notice that by adding a number to 0, we get the same number and in multiplication we also get the same number by multiplying by 1. They are called their identity.  


Therefore, for both addition and multiplication we have numbers, inverses and an identity.

Now we are ready to learn the textbook definition of a group.

 Group Definition
  • Set of elements G
  • Operation: *
  • Closed under operation
  • Inverses: x*x-1 exists for all x
         x*x-1=e(Identity)
  • Identity: y*e=e*y=y
  • Associativity:

    (a*b)*c=a*(b*c)
We have elements which is more abstract than numbers, we have an operation(in our example of numbers it was +, x ). Note it is closed under an operation meaning the result of the operation is also an element of the group. Each element has an inverse. If we combine an element with its inverse, we get the identity element. We also have associativity because without it, we cannot solve the simplest of equations. Groups do not have to be commutative but when it is, we call it a commutative group or an Abelian group.

x*y≠y*x (Non commutative group)
x*y=y*x (Abelian group)

There you have it! Group theory is study of groups and is among the primary pillars of Abstract algebra.

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