let s be a semigroup. prove that s is a group if and only if the following conditions hold:
1. s has a left identity—there exists an element e ∈ s such that ea = a for all a ∈ s;
2. each element of s has a left inverse—for each a ∈ s, there exists an element a -1 ∈ s such that a -1a = e.

Answer with Step-by-step explanation:

Let S be  a semi-group.

Semi group: It is that set of elements which satisfied closed property and associative property.

We have to prove that S is  a group if and only if the following condition hold'

1.S has  left identity -There exist an element such that ea=a for all

2.Each element of S has a left inverse - for each , there exist an element such that

1.If there exist an element such that ea=a for all

Then , identity exist in S.

2.If there exist left inverse for each

Then, inverse  exist for every element in S.

All properties of group satisfied .Hence, S is a group.

Conversely, if S is a group

Then, identity exist and inverse of every element exist in S.

Identity exist: ea=ae=a

Inverse exist:

Hence, S has  a left identity for every element

and each element of S has a left  inverse .

Hence, proved.