We establish several addition theorems on finite abelian groups by employing a group as a useful tool. Among several results the following is proved. Let p be a prime, and let G = Z(p)(sic) x...xZ(p)(sic) with 1 less than or equal to e(1) less than or equal to...less than or equal to e(n). Put w={1/(p(en) - 1)) x Sigma(sic)(n)(p(en)-1). Then, for any t greater than or equal to(p(en)-1) log w + p(en) - 2 linear bases B-1,..,B-t of G their union (with repetitions) U-sic(t) B-t forms an additive basis of G. (C) 1997 Academic Press
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