Question

let G be a group then for a, b, c€G,ab=ac ,b=c (left cancellation law) and bc=ca,b=c (right cancellation law)
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Answer 1

Suppose a⋅b=a⋅c

Let a−1 be the inverse element of a in G (s.t. a−1⋅a=a⋅a−1=e where e is the identity element), which must exist by the axioms of groups. Now consider : a−1⋅(a⋅b)=a−1⋅(a⋅c)

By associativity, we have

(a−1⋅a)⋅b=(a−1⋅a)⋅c

By the definition of inverse, we have

e⋅b=e⋅c

where e is the identity element (s.t. e⋅x=x⋅e=x for all x∈G). By the definition of the identity element.

=> b = c