A note on the exterior centralizers and the probability of commuting pairs

We show two structural results for the subclass of $FC^\wedge$-groups, belonging to the well-known class of groups with finite conjugacy classes (i.e., $FC$-groups). We focus on linear $FC^\wedge$-groups and on the notion of exterior centralizer in the nonabelian exterior square of groups. Final comments show the relevance of $FC^\wedge$-groups and $FC$-groups in the theory of commuting probabilities of topological groups.