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.
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Expert's answer
2013-02-06T07:38:03-0500
Let a ∈Δ(G)be such that a' has finite order in the quotient group Q = Δ(G)/Δ+(G). Then an∈Δ+(G) for some n ≥ 1, and hence (an)m= 1 for some 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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