This paper studies the group G(A X B) of (homotopy classes of) self-homotopy equivalences of a product A X B of two connected CW homotopy associative ϋf-spaces A and B. It establishes the existence of an exact sequence of multiplicative groups 1-> [A A B, A X B]-> G{A X B)-> GL(2, A TJ)-» 1 provided that io[A x B, A X B]oqo[A A B, A X B] = 0, where A Λ B is the cofibration induced by the inclusion i: A\/ B-> Ax B of the sum into the product. The entry GL(2, ΛJJ) is the group of invertible matrices rh Λ-fh A A h AB \ KPΊJ)-7 7 \JlBA hβB/ with entries h ΣJ in the homotopy sets Λu = [I, J] for I, J = A, B, where matrix multiplication is defined by (huXku) = (h IA ok A i + hiB°kBj) in terms of composition o and the operation + in the homotopy sets [/,/], and where the multiplicative unit is
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