Question

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 1

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
`e\in S` such that ea=a for all
`a\in S`

2.Each element of S has a left inverse - for each
`a\inS`, there exist an element
`a^(-1)\in S` such that
`a^(-1)a=e`

1.If there exist an element
`e\in S` such that ea=a for all
`a\in S`

Then , identity exist in S.

2.If there exist left inverse for each
`a\inS`

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:
`aa^(-1)=a^(-1)a=e`

Hence, S has a left identity for every element
`a\in S`

and each element of S has a left inverse .

Hence, proved.