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In this paper, using an algebraic approach, it is intended to show that theGoldbach's and Twin primes conjectures are true, building, for each $m>2$, anisomorphism between posets. One of the posets is the set of coprimes less than$m$, while the other is endowed with an operation that grants it an abeliangroup structure. Special features of this operation are demonstrated in thedocument, which allow characterizing the even numbers, as if they were theirfingerprint; furthermore, such an operation locates, in a natural way, thepairs that satisfy the conjecture. Moreover, an algorithm that generates pairsof numbers that satisfy the conjecture is presented, and examples of some ofthe beautiful symmetries of the orbits of cyclic subgroups are shown, for theproposed abelian group.

Is it possible that the Goldbach's and Twins primes conjectures are true with an algebraic approach?