AbstractWe investigate the provability of some properties of abelian groups and quadratic residues in variants of bounded arithmetic. Specifically, we show that the structure theorem for finite abelian groups is provable in S22 + iWPHP(Σ1b), and use it to derive Fermat's little theorem and Euler's criterion for the Legendre symbol in S22 + iWPHP(PV) extended by the pigeonhole principle PHP(PV). We prove the quadratic reciprocity theorem (including the supplementary laws) in the arithmetic theories T20 + Count2(PV) and I Δ0 + Count2(Δ0) with modulo‐2 counting principles (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)