Abstract: Basing on Chebyshev inequality and on the computation of average concentration of all related subsets, the authors put forward and proved the following lemma, theorem, and the four corollaries: Lemma 1 There exists at least one of the sets of generalized prime-twins (namely one subset of the set of 2-primes group), which is an infinite set. Theorem 1 All the sets of generalized prime-twins or infinitely many ones among these sets are infinite sets. Corollary 1 There exists at least one of the sets of primes-triplet (namely one subset of the set of 3-primes-group), which is an infinite set. Corollary 2 All the sets of primes-triplet or infinitely many ones among these sets are infinite sets. Corollary 3 There exists at least one of the sets of h-primes-tuplet (namely one subset of the set of h-primes-group) which is an infinite set, where h is an inte ger≥2. Corollary 4 All the sets of h-primes-tuplet or infinitely many ones among these sets are infinite sets, where h is an integer≥2.