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let (G,*) be a semigroup with the property that for any two elements a,b belongs to G , the equations:

a*x=b , y*a=b

have solutions x,y in G , verify that (G,*) forms a group.

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my attempt

first one ,

* is associative as (G,*) is a semigroup

assume a=b ,then , a*x1=a , y1*a=a

so x is a left identity and y is the right one .

let c belongs to G then

c*x2=c , y2*c=c

if I can prove that x1 = x2 then x1 is the identity , but I have no idea about the way to do

that ! ,

if I could prove that e is an identity element in G then it's easy to proof that there is an inverse for all a belongs to G :

let b = e then

a*x=e , and x belongs to G " that is mentioned in the question " so the solotion is the inverse .

so the problem is in the identity

the solution shouldn't use any theorem related with groups because the exercise is after the defenition of group and it's examples and no theorem has explained

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# Homework Help: Prove that (G,*) is a group .

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