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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 &isin;&Delta;(G)be such that a&#039; 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&#039; = 1. This showsthat Q is torsion-free. Since &Delta;(G) is an f.c. group, so is Q.But then, Q must be abelian.

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