It is known that if a group G has an abelian subgroup of finite index n, then it contains an abelian characteristic subgroup A of index at most n(n). The aim of this paper is to improve this bound by showing that the characteristic subgroup A can be chosen of index at most n(2). Examples prove that this bound is the best possible. Our main result is obtained as an application of a general method for the construction of large characteristic subgroups.
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