A group G is said to be factorized into subsets A1, A2, . . . , As ? G if every element g in G can be uniquely represented as g = g1g2 . . . gs, where gi ∈ Ai, i = 1, 2, . . . , s. We consider the following conjecture: for every finite group G and every factorization n = ab of its order, there is a factorization G = AB with |A| = a and |B| = b. We show that a minimal counterexample to this conjecture must be a nonabelian simple group and prove the conjecture for every finite group the nonabelian composition factors of which have orders less than 10 000.
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