Higman was born in Louth, Lincolnshire and attended Sutton High School, Plymouth, winning a scholarship to Balliol College, Oxford. In 1939 he co-founded The Invariant Society, the student mathematics society, and earned his DPhil from the University of Oxford in 1941. His thesis, The units of group-rings, was written under the direction of J. H. C. Whitehead. From 1960 to 1984 he was the Waynflete Professor of Pure Mathematics at Magdalen College, Oxford.
He was also a local preacher in the Oxford Circuit of the Methodist Church. During the Second World War he was a conscientious objector, working at the Meteorological Office in Northern Ireland and Gibraltar.
He died in Oxford.
- Higman–Sims group, named after Donald G. Higman, but studied also by Graham Higman.
- Higman's embedding theorem
- Feit-Higman theorem
- Higman group
- Higman's lemma
- HNN extension
- Hall–Higman theorem
- on YouTube
- Graham Higman (1940), The units of group-rings, Proceedings of the London Mathematical Society, (2) 46: 231–248.
- Walter Feit and Graham Higman (1964), The nonexistence of certain generalized polygons., J. Algebra, 1: 114–131
- Graham Higman (1966) Odd characterisations of finite simple groups, U. of Michigan Press
- *Graham Higman (1974), Finitely presented infinite simple groups, Notes on Pure Mathematics, 8, Department of Pure Mathematics, Department of Mathematics, I.A.S. Australian National University, Canberra, ISBN 978-0-7081-0300-5, MR 0376874
- Graham Higman and Elizabeth Scott (1988), Existentially closed groups, LMS Monographs, Clarendon Press, Oxford
- Collins, Michael (2008-05-08). "Professor Graham Higman: Leading group theorist". Obituaries. The Independent. Retrieved 2008-10-14.
- The Early History of the Invariant Society by Robin Wilson, printed in The Invariant (2010), Ben Hoskin
- Hickin, Kenneth (1990). "Review: Existentially closed groups by Graham Higman and Elizabeth Scott" (PDF). Bull. Amer. Math. Soc. (N.S.). 23 (1): 242–249. doi:10.1090/s0273-0979-1990-15943-9.