It is shown that the set of integer solutions of a single Diophantine equation, or of several simultaneous linear Diophantine equations, in an arbitrary number of variables, say $n$, is a group with respect to a suitably defined binary operation. Further, the aforesaid group is the direct sum of $n$ cyclic groups.
"Integer Solutions of Linear Diophantine Equations Form a Group." Missouri J. Math. Sci. 18 (2) 135 - 141, May 2006. https://doi.org/10.35834/2006/1802135