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)/Δ+(G) is torsion-free abelian.

Answer 1

Let a &isin;&Delta;(G)be such that a' has finite order in the quotient group Q = &Delta;(G)/&Delta;⁺(G). Then aⁿ &isin;&Delta;⁺(G) for some n &ge; 1, and hence (aⁿ)^m= 1 for some m &ge; 1. But then a &isin;&Delta;⁺(G), and so a' = 1. This showsthat Q is torsion-free. Since &Delta;(G) is an f.c. group, so is Q.But then, Q must be abelian.