Integer addition is associative, even for fixed-bitwidth integers (which constitute "the integers modulo 2^n" for some bitwidth n; e.g. 8-bit bytes under addition form the group of integers modulo 2^8 (i.e., integers modulo 256)).

Note that both the full integers as well as the integers modulo n are groups under addition. As groups are more restrictive than semigroups, all groups are also semigroups (groups are semigroups that also have an identity element and where every element is invertible).

Integer multiplication also forms a semigroup (and in fact a monoid, due to identity), but not a full group due to zero not having an inverse. So, it's usually said that "the non-zero integers" form a group.