Answer to Question #23547 in Abstract Algebra for Hym@n B@ss

Question

Let G be an f. c. group. Show that [G,G] is torsion

Answer 1

If G is f. c., consider any g∈[G,G]. There exists a finitely generated subgroup H ⊆ G such that g ∈[H,H]. Since H is also f. c., the order of g divides|[H,H]| < ∞.