Group

A monoid (S,*) with identity element ’e’ is called a group if to each element a∈S, there exist an element b∈S such that (a*b)=(b*a)=e then ‘b’ is called inverse of a (denoted as a⁻¹). b=a⁻¹ and a=b⁻¹

Example #

• (Z,+): yes
  • It is a monoid because it has the properties of closure, associativity, and has an identity element ‘0’.
  • a+b=e=0 ∀a∈Z. b=-a∈Z. ‘a’ has an inverse b=-a
• (R,⋅): no
  • It is a monoid because it has the properties of closure, associativity and has an identity element ‘1’
  • a⋅b=1 ∀a∈R b=1/a but for a=0, an inverse ‘b’ doesn’t exist under multiplication belonging to the set R.
  • But (R*,⋅) is a group when we exclude 0 from R.