Convolutions of slowly oscillating functions
Jack P. Tull · Proceedings of the American Mathematical Society · 1962
CONVOLUTIONS OF SLOWLY OSCILLATING FUNCTIONS 225r= 23p r(P) Men G is quotient divisible.Proof.For each p, G/pG is a direct sum of cyclic groups of order p: G/pG= 23(¿,P)6/(p) Z(x(i, p)+pG) for some elements x(i, p)EG.Evidently, r(p) = \ I(p)\.Since rank ¿>G = rank G = r for each p, we may apply Theorem 2.1 with N= {p\p is a prime}, Ap = pG for each PEN and BP = G for each pEN, and obtain a free group FQG such that F+pG = G for each p. Applying Lemma 3.3, with S any basis of F, we conclude that G is quotient divisible.Corollary 3.5.4«y torsion free group of infinite rank is quotient divisible.Proof.£Pr(£) g £pr = fcV = r.