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The paper is concerned with the subsets  I  ⊂ {0, ...,  d  – 1} for which gcd( n  –  m ,  d ) ≠ 1 for any  n ,  m  ∈  I . Such subsets are called sets of nontrivial differences. Let  d  &gt; 1 and  d  1  be the least prime divisor of  d . We prove that the largest cardinality of a set of nontrivial differences is  d / d  1 . Sets of nontrivial differences in which not all differences of elements are multiples of the same prime factor  d  are called nonelementary. Let  t  be the number of prime factors of  d . We show that there are no nonelementary sets for  t  ⩽ 2. It is shown that a minimal nonelementary set may have arbitrary order in the interval3  ,    t   ¯$\overline {3,\;t}$    . The largest cardinality of nonelementary sets is estimated from below and above.

Cardinality of subsets of the residue group with nonunit differences of elements