Can monoids of endomorphisms of nonisomorphic groups be isomorphic ?



Monoids of endomorphisms of nonisomorphic groups
For any prime $p$, the endomorphism monoid of $\mathbb{Z}[\frac{1}{p}]$ is a commutative monoid with zero whose submonoid of nonzero elements is the direct sum of a cyclic group of order two (generated by multiplication by $-1$), an infinite cyclic group (generated by multiplication by $p$), and a free commutative monoid on countably many generators (multiplication by other primes).


But different primes give nonisomorphic groups.


