### 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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