Uniqueness of identity in a group : Group theory
Theorem : In a group , the identity element is unique : proof : Suppose G is a group and e be the identity element in G. Now we prove e is only the identity element in G. In contrary assume e' is another identity element in G. To prove the identity element is unique, we have to prove e = e'. Since e is identity in G, we have e a = a = a e ∀ a 𝞊 G. In particular , since e' 𝞊 G, we have e e' = e' = e' e .............I Since e' is identity in G, we have e' a = a = a e' ∀ a 𝞊 G. ...