HOUSTON JOURNAL OF MATHEMATICS Volume 17, No. 3, 1991 ON THE NEAR FRATTINI SUBGROUP OF THE AMALGAMATED FREE PRODUCT OF FINITELY GENERATED ABELIAN GROUPS MOHAMMAD K. AZARIAN Abstract. Let (3 be the amalgamated free product of finitely many finitely generated abelJan groups Gi(1 • i • n) with amalgamated subgroup H, and let •b(G) be the near Frattini subgroup of (3. We prove that •b(G) = Tor H, where Tor H is the torsion subgroup of H. Alice Whittemore [5, Theorem 2, p. 324], has shown that the Frattini subgroup of the free product of finitely many finitely generated abe]Jan groups is contained in the torsionsubgroup of the amalgamated subgroup. In [1] the author proved the exact analog of Whittemore's Theorem for the near Frattini subgroup: Theorem 1 [1, Theorem 3, p. 527]. If G is the freeproductof 1initely many i]nitely generated abelian groups with amalgamated subgroup H, then•b(G) < Tot H. Our aim in this paper is to improveTheorem 1. That is, we want to show that if G and H are as in Theorem 1, then the nearFrattini subgroup of G coincides with the torsion subgroup of H. Our notation will be standard. For amalgamated free products of groups we accept B. H. Neumann's viewpoint [3], and we useJ. B. Riles paper[4] for the definitions to follow. An element g of a group G is a near generator of G if there is asubset S of G such that G :< S > I = oo, but IG :< g, S > I < oo. Hence, an element g of G is a non-near generator of G precisely whenS C_ G and ]G :< g,S > ] < oo always imply that ]G :< S > ] < oo. A subgroup M ofagroup G is nearly maximal in G if]G' M] = oo,but IG' N] < oo,whenever M < N < G. That is, M is maximal with respectto being of infinite index in G. The set of all non 425