Richard Neil Lyons (born January 22, 1945 in New York City, New York) is an American mathematician, specializing in finite group theory.
Lyons received his PhD in 1970 at the University of Chicago under John Griggs Thompson with a thesis entitled Characterizations of Some Finite Simple Groups with Small 2-Rank. Since 1972 he has been a professor at Rutgers University.
With Daniel Gorenstein and Ronald Solomon he wrote a six volume series on the classification program for finite simple groups, a program in which the three of them were major participants. He discovered a sporadic group which Charles Sims constructed and called the Lyons group Ly.
In 2012 he shared the Leroy P. Steele Prize for Mathematical Exposition, awarded by the American Mathematical Society, with Michael Aschbacher, Stephen D. Smith, and Ronald Solomon. In 2013 he became a fellow of the American Mathematical Society "for contributions to the classification of the finite simple groups, including the discovery of one of the 26 sporadic finite simple groups.".
- with Gorenstein: The local structure of finite groups of characteristic 2 type, American Mathematical Society, 1983
- with Daniel Gorenstein, Ronald Solomon: The classification of the finite simple groups, American Mathematical Society, 6 Vols., 1994–2005
- with Michael Aschbacher, Stephen D. Smith, Ronald Solomon: The classification of finite simple groups: Groups of characteristic 2 type, American Mathematical Society, 2011