La bazele algebrei (engleza)

Title:

A Remark on the Existence and Uniqueness of Solutions of an Equation

in a Group

Abstract:

It is shown that for the equation a⋅x=b in a group (G,⋅), the

existence of a solution automatically implies its uniqueness.

Observation / Proof:

In standard algebra textbooks, the proof of the uniqueness of the solution

to the equation a⋅x = b (a,b ∈ G, G group)is often divided into two steps:

first, existence is shown, and then uniqueness is proved. In my view, this

separation can be confusing, because the solution already found guarantees

uniqueness due to the properties of equality(equivalence relation).

a⋅x=b ⟺ a'⋅(a⋅x)=a'⋅b (multiply both sides on the left by a')

⟺ (a'⋅a)⋅x=a'⋅b (associativity)

⟺ e⋅x=a'⋅b (a'⋅a = e)

⟺ x=a'⋅b (e is the identity element)

Thus, a solution exists and it is unique: the equality “=” ensures that

there cannot be any other element x satisfying the equation, without

separating existence and uniqueness into two steps.