Question

Let "G" be a Group and let "a" and "b" element of "G" then the equation ax = b has a unique solution in "G".​

Answer 1

Answer:

I think we only need ax=b or ya=b have solutions in G, I'll prove it.

Proof:

(I)

G is a group ⟹ a−1ax=a−1b ⟹ x=a−1b ⟹ ax=b has solutions in G

(II)

(A) ax=b has solutions in G ⟹ ax=a has solutions in G ⟹ there exists an identity element e∈G such that ae=ea=a

(B) ax=b has solutions in G ⟹ ax=e has solutions in G ⟹ there exists an inverse element a−1∈G such that ae=ea=a

Hence, ax=b has solutions in the semigroup G (for all a,b∈G) if and only if G is a group.

Correct?

Reference: Fraleigh p. 49 Question 4.39 in A First Course in Abstract Algebrab