Using a sample of (225.2±2.8)×106 J/ψ events collected with the BESIII detector, we present results of a study of J/ψ→γΛΛˉ and report the first observation of the isospin violating decay J/ψ→ΛΣˉ0+c.c., in which Σˉ0 decays to γΛˉ. The measured branching fractions are B(J/ψ→ΛˉΣ0) = (1.46±0.11±0.12)×10−5 and B(J/ψ→ΛΣ0ˉ) = (1.37±0.12±0.11)×10−5. We search for Λ(1520)→γΛ decay, and find no evident signal, and an upper limit for the product branching fraction B(J/ψ→ΛΛˉ(1520)+c.c.)×B(Λ(1520)→γΛ)<4.1×10−6 is set at the 90% confidence level. We also report the observation of ηc→ΛΛˉ in J/ψ→γηc, ηc→ΛΛˉ and measure the branching fraction B(ηc→ΛΛˉ)=(1.16±0.12(stat)±0.19(syst)±0.28(PDG))×10−3.