# Math Help - Semigroup with left id and inverse

1. ## Semigroup with left id and inverse

Suppose that S is a semigroup with left identity and left inverse, prove that it is a group.

Well, since it is a semigroup, S already has the associativity property, but how do I show it has the right id and right inverse?

Thank you

let $e$ be a left identity and $x \in S.$ so $\exists y \in S: \ yx=e.$ also $\exists z \in S: \ zy=e.$ thus $e=zy=z(ey)=z(yx)y=(zy)xy=e(xy)=xy.$
so $y$ is also a right inverse of $x.$ finally we have: $xe=x(yx)=(xy)x=ex=x,$ i.e. $e$ is a right identity as well. Q.E.D.