Question

A monoid is called a group if _______

(a) (a*a)=a=(a+c)

(b) (a*c)=(a+c)

(c) (a+c)=a

(d) (a*c)=(c*a)=e

The question was posed to me during an interview.

I need to ask this question from Group Theory in portion Groups of Discrete Mathematics

Answer 1

Right option is (d) (a*c)=(c*a)=e

Easy explanation: A monoid(B,*) is called Group if to each element there exists an element c such that (a*c)=(c*a)=e. Here e is called an identity element and c is defined as the inverse of the corresponding element.