Higman was born in Louth, Lincolnshire and attended Sutton's Secondary 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
- Graham Higman (1940), "The units of group-rings", Proceedings of the London Mathematical Society, (2) 46: 231–248.
- 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
- O'Connor, John J.; Robertson, Edmund F., "Graham Higman", MacTutor History of Mathematics archive, University of St Andrews.
- Graham Higman at the Mathematics Genealogy Project
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