If G is locally finite of exponent n then χ(G) is locally finite of finite n-bounded exponent, plus finiteness criteria for the commutator subgroup D(G).
Shumyatsky, Elements of prime power order in residually finite groups , Int
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Finiteness conditions for the weak commutativity construction
If G is locally finite of exponent n then χ(G) is locally finite of finite n-bounded exponent, plus finiteness criteria for the commutator subgroup D(G).