On cardinalities in quotients of inverse limits of groups

by Grossberg and Shelah. [GrSh:302a]
Math Japonica, 1998
Let lambda be aleph_0 or a strong limit of cofinality aleph_0 . Suppose that < G_m, pi_{m,n} : m <= n< omega > and < H_m, pi^t_{m,n} : m <= n< omega > are projective systems of groups of cardinality less than lambda and suppose that for every n< omega there is a homorphism sigma :H_n-> G_n such that all the diagrams commute. If for every mu < lambda there exists < f_i in G_{omega} : i< mu > such that i not= j ===> f_if_j^{-1} not in sigma_{omega}(H_{omega}) then there exists < f_i in G_{omega} : i < 2^{lambda}> such that i not= j ===> f_if_j^{-1} not in sigma_{omega}(H_{omega}) .


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