  In this paper, using an algebraic approach, it is intended to show that the
Goldbach's and Twin primes conjectures are true, building, for each $m>2$, an
isomorphism 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 abelian
group structure. Special features of this operation are demonstrated in the
document, which allow characterizing the even numbers, as if they were their
fingerprint; furthermore, such an operation locates, in a natural way, the
pairs that satisfy the conjecture. Moreover, an algorithm that generates pairs
of numbers that satisfy the conjecture is presented, and examples of some of
the beautiful symmetries of the orbits of cyclic subgroups are shown, for the
proposed abelian group.
