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Group. v(G) < ~ if if and and only G D divisible torsion G == D~ H, with D aa divisible torsiongroup, group, and and HH aa torsion free free group group with v(H) < ~. v(G) << ~. 9. 9. Theorem D divisible torsion Conversely, suppose that G aa divisible torsion group, group, and Conversely, suppose G == D(f) H, D H v(H) = nn < ~. Let R R be an an associative associative ring H aa torsion free free group group with with v(HJ be ideal in R, and (RIO)+~ H. Hence G. )u is an an ideal with H+ = G. (R,D)n+l R2n+2 2n+ 2 c o 2 == 0 by Theorem (R/D)n+l == 0, or Rn+l c=D.

Therefore G with If G == (x (x1) ® (x2), I ) (f) (x J, lx; I = n i = 1,2. If , 1x11 = 1 1 2 (n1, Otherwise let let pp be (n = 1, 1, then G is cyclic. cyclic. Otherwise be aa prime n2) = G primedivisor divisor of 1 , n2) 3) 1 m. p', (n11n2). G == (y (y1) (Y2) (n H, with with IY; 1 == p 1 , i = 1,2, and 1 ,n 2). Then G 1) (t)® (y 2 )@®H, < m2. 1. 2. 1. m -1 The products Y;·Yj = induce an an associative associative ring The y2 for i,j i,j==1,2, 1,2, induce = pp 2 y2 1 1 2 ~ 0. Therefore R =

4, ••• ). The 3) ==(4,4, 2 ) =={2,2, for u-i i+j = e. •e. •e. 1 J 1 3 ~ 3 = 1 0 otherwise induce an anassociative associative ring ring structure R R G with e~ = e3 induce on G e 3 ~ 0. 4. •e. (. e1) = e4. S4 N(G) = ) (e 1 ·e 1 ) • (e ·e = e . 6. 1 1 4 Theoremare areprecisely precisely attained. attained. In this thisexample example the tne bounds bounds of of Webb's Webb's Theorem another such see [72]. [72]. 8: For G satisfying v(G) A group G N(G) == ~. v(G) << N(G) be aa rank one torsion torsion free group iI == 11,2, ,2, let G.