Here I explain the proof that in ZF, the axiom of choice (AC) is equivalent to every nonempty set having group structure (GS). I first learned of the nontrivial direction of this argument in this MathOverflow post and as far as I know first appeared in “Some new algebraic equivalents of the axiom of choice” by A. Hajnal and A. Kertész in Publ. Math. Debrecen 19 (1972), pp. 339-340.