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.
In particular , since e 𝞊 G, we have e' e = e = e e' .............II
From I and II , we have e = e'
∴ e is only the identity element in G.
Thus, in a group, the identity element is unique.
*** Hence the proof ***
* uniqueness of inverse element in a group
* Is aob = ab is a binary operation on N?
* Is aob = a is associative in S?
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#algebra #syllabus #semi #groupoid #monoid #abelian #commutative
#Theorem : #In #a #group #, #the #identity #element #is #unique #:
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