At the age of seventeen, Miller began school-teaching to raise funds for higher education. In 1882, he entered Franklin and Marshall Academy, and progressed to
Muhlenberg College in 1884. He received his B.A. in 1887 and M.A. in 1890. While a graduate student, Miller was Principal of schools in
Greeley, Kansas and then professor of mathematics as
Eureka College in Eureka, Illinois. He corresponded with
Cumberland University in Lebanon, Tennessee for his Ph.D. in 1892. He then joined
Frank Nelson Cole at the
University of Michigan and began to study
groups. In 1895, he traveled to Europe, where he heard
Sophus Lie lecture at Leipzig and
Camille Jordan at Paris. In 1897, he traveled to
Cornell University, where he worked as an assistant professor, and, in 1901, went to
Stanford University, where he worked as associate professor. In 1906, he went to the
University of Illinois, where he taught until his retirement in 1931. Miller helped in the enumeration of
finite groups of degree 8, 9, and 10.
Arthur Cayley had listed 198 groups of degree 8 in 1891, and Miller found two more making the total 200 in 1893. Camille Jordan had given a list for degree 9 in 1872, re-examined by Cole, and brought up to 258 groups by Miller. In 1894 Miller produced a list of 294 intransitive groups of degree 10. In consequence, the Academy of Science of Cracow awarded a prize and "Miller came to prominence in the mathematical world abruptly." and gave a
plenary address at the International Congress of Mathematicians in 1924 in Toronto. Miller's
Collected Works were edited by
Henry Roy Brahana and published by
University of Illinois Press, the first two volumes appearing in 1935 and 1939. The final three volumes were published in 1946, 1955, and 1959. His doctoral students include
H. L. Rietz. ==Publications==