Solomon studied as an undergraduate at Queens College and received a PhD in 1971 at Yale University under Walter Feit with a thesis entitled Finite Groups with Sylow 2-Subgroups of the Type of the Alternating Group on Twelve Letters. In 1972 he began his participation in the classification program for finite simple groups, after hearing a lecture by Daniel Gorenstein. He was for two years an instructor at the University of Chicago and the academic year 1974–1975 at Rutgers University, before he became a professor at Ohio State University, where he has remained. In 2006 he received the Levi L. Conant Prize and in 2012 the Leroy P. Steele Prize. In 2012 he was elected a Fellow of the American Mathematical Society.
With Gorenstein and Richard Lyons he wrote a series of six volumes on the classification program for finite simple groups in the context of a project for the examination of the proof.