# Answer to Question #23551 in Abstract Algebra for Hym@n B@ss

Question #23551

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)/Δ+(G) is torsion-free abelian.

Expert's answer

Let

*a**∈**Δ*(*G*)be such that*a'*has finite order in the quotient group*Q*=*Δ*(*G*)*/**Δ*^{+}(*G*). Then*a*^{n}*∈**Δ*^{+}(*G*) for some*n ≥*1, and hence (*a*)^{n}*= 1 for some*^{m}*m ≥*1. But then*a**∈**Δ*^{+}(*G*), and so*a'*= 1. This showsthat*Q*is torsion-free. Since*Δ*(*G*) is an f.c. group, so is*Q*.But then,*Q*must be abelian.
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