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Using (1.0087 ± 0.0044) × 1010 J/ψ events collected with the BESIII detector at the BEPCII storage ring, the reactions Σ+n → ¢pand Σ+n → Σ0pare studied, where the Σ+ baryon is produced in the process J/ψ → Σ+─− and the neutron is a component of the 9Be, 12C, and 197Au nuclei in the beam pipe. Clear signals of these two reactions are observed for the first time. Their cross sections are measured to be σ(Σ+ + 9Be → ¢+ p+ X) = (45.2 ± 12.1stat ± 7.2sys) mb and σ(Σ+ + 9Be → Σ0 + p+ X) = (29.8 ± 9.7stat ± 6.9sys) mb for a Σ+ average momentum of 0.992 GeV/c, within a range of ±0.015 GeV/c, where X represents the residual nucleus. This is the first study of Σ+-nucleon scattering at an electron-positron collider. The measured results serve as a significant element in enhancing the comprehensive understanding of hyperon-nucleon interactions, and also constitute an important factor in resolving the “hyperon puzzle” of neutron stars.