Graham Higman was born in Louth, Lincolnshire, on 19 January 1917, the second son of Joseph and Susan Mary Higman (née Ellis). His father was a United Methodist minister, of Bible Christian stock, and the family moved from Lincolnshire to London (1918-24), and then to Long Eaton (1924-29) and to Plymouth, where Graham was educated at Sutton High School for Boys. In 1934 he won a scholarship to read Natural Sciences at Balliol College, Oxford, but chose to read Mathematics instead, in order to distinguish himself from his elder brother Bryan, who had studied Chemistry at Balliol. Specialising in Pure Mathematics, Higman took a First in 1938, stayed in Oxford to complete a doctorate, and then moved to Cambridge for a year.
While his brother served in the RAF during the Second World War, Graham was a conscientious objector, working for the Meteorological Office. In 1946 he was appointed as a lecturer at Manchester University, returning to Oxford in 1955 as Reader in Mathematics. He was elected to a Fellowship of the Royal Society in 1958 and occupied the Waynflete Chair of Mathematics from 1960 until his retirement in 1984, moving on appointment from Balliol to Magdalen College. His work, principally on group theory, was influential, and, although formidable in manner and sometimes eccentric in appearance, as a teacher he was generous with time, assistance and credit.
He married Ivah Treleaven in 1941, and they had five sons and a daughter. Both Graham and Ivah were active members of Wesley Memorial Church, Oxford, and voluntary workers with the Samaritans. Higman became a fully accredited Local Preacher in 1936, continuing to preach until 2001. He died in Oxford on 8 April 2008.