[Math] Understanding the Group Structure

group-theory

A Group is an algebraic structure consisting of a set of elements together with an operation that combines any two elements to form a third element. The operation satisfies four conditions

  1. Closure
  2. Associativity
  3. Existence of Identity
  4. Existence of Inverse

Intuitively I understand the purpose of Closure and Associativity Property. But I'm not getting the intuition behind Identity and Inverse. Whats the purpose of having these elements in a group.

I searched everywhere and find only common definition which are straight forward. But nobody discuss why it's important. Can anybody explain it to me.

Best Answer

The existence of an inverse (and of a neutral element or identity with respect to which it is defined) is the fact that guarantees that we can solve an equation of the form: $a*x=b$ and find $x=a^{-1}b$.

This seems a good reason to deserve a special name ( and attention) to such a structure.

In other words: the existence of a neutral element $e$ and an inverse $a^{-1}$ make a group the simpler structure in which we can solve the equation:

$$ a*x=b \Rightarrow (a^{-1}*a)*x=a^{-1}*b \Rightarrow e*x=a^{-1}*b \Rightarrow x=a^{-1}*b $$

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