11.56 (1990)

a) Does every infinite residually finite group contain an infinite abelian subgroup? This is equivalent to the following: does every infinite residually finite group contain a non-identity element with an infinite centralizer?

By a famous theorem of Shunkov a torsion group with an involution having a finite centralizer is a virtually soluble group. Therefore we may assume that in our group all elements have odd order. One should start, perhaps, with the following:

b) Does every infinite residually $p$-group contain an infinite abelian subgroup?