Yes, for example in Osin's infinite group with 2 conjugacy classes every maximal cyclic proper subgroup is big. Of course if you do not care about the number of generators, you can consider the (much easier) infinitely generated group constructed by Higman-Neumann-Neumann where all non-identity elements are conjugate. There also every proper subgroup is big.
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Yes, for example in Osin's infinite group with 2 conjugacy classes every maximal cyclic subgroup is big. |
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