Question

For any group G, let
Δ(G) = {g ∈ G: [G : CG(g)] < ∞}, and Δ+(G) = {g ∈ Δ(G) : g has finite order}.
Show that Δ+(G) is a characteristic subgroup of G and that Δ+(G) is the union of all finite normal subgroups of G.