national academy of sciences george aBram miller 1863—1951 A Biographical Memoir by h. r. Brahana Any opinions expressed in this memoir are those of the author(s) and do not necessarily reflect the views of the National Academy of Sciences. Biographical Memoir Copyright 1957 national aCademy of sCienCes washington d.C. GEORGE ABRAM MILLER 1863-1951 BY H.R. BRAHANA GEORGE ABRAM MILLER was born near Lynville, Pennsylvania, on July 31, 1863, the year the National Academy of Sciences was incorporated. He was elected to membership in the Academy in 1921. He published two papers in the first volume of the Proceedings of the Academy in 1915; he contributed five or six papers to each volume of the Proceedings for many years around the time of his retirement in 1931; and his last two mathematical papers appeared in Volume 32 of the Proceedings in 1946. He died at Urbana, Illinois, on February 10, 1951. Miller came into prominence in the mathematical world abruptly in 1894-1895 when he completed the determination of the substi- tution groups of degrees eight and nine. The groups of degree eight had been listed in 1891 by Cayley, who at that time was referred to as the greatest English mathematician since Newton. Cayley's list had been corrected by F. N. Cole, and Miller started his study of groups after his association with Cole. Miller redetermined the groups and brought the number to 200 including two groups that his predecessors had missed. Camille Jordan, one of the foremost French mathematicians and a specialist in the theory of groups, had published in 1872 a list of the primitive groups of degree nine and Cole had given a list of all the groups of degree nine in 1893; Miller determined the 258 groups of this degree, adding one group to Jordan's list and two to Cole's. In the following year he published his own list of 994 intransitive groups of degree ten. These lists 258 BIOGRAPHICAL MEMOIRS have stood since that time. In 1900 the Academy of Sciences of Cracow awarded him a prize for his work on the groups of degree ten, a prize which had been standing since 1886. This was the first award to an American for work in pure mathematics. The mathematical world which Miller entered in 1894 was largely a European world, and mathematics was just getting a foothold in America. Although mathematics is an old subject and advanced vigorously during the two preceding centuries, there had been no participation by Americans in the advancement. The first notable American contribution was a memoir on "Linear Associative Alge- bras" by Benjamin Peirce of Harvard. This was read before the National Academy of Sciences in Washington in 1870, although at that time there was no organized mathematical group to which to present it and no American mathematical periodical in which it could be published. It was published later in the American Journal of Mathematics. The Johns Hopkins University, established in 1876, is considered to have started the first school of mathematics in this country, headed by the English mathematician, J. J. Sylvester. Harvard, Yale, and Princeton were offering the degree of Doctor of Philosophy in mathematics, but Harvard Fellows were going to Germany to study mathematics. Some returned to Harvard for their degrees while others took their degrees abroad. Cole got his degree at Harvard in 1886 after two years in Germany; Bocher got his degree in 1891 at Gottingen after some years there as a Harvard Fellow. Clark University, established in 1889, had Bolza, a German, on its staff. The University of Chicago, established in 1893, had E. H. Moore and two Europeans, Bolza and Maschke, in its mathe- matics department; Moore had his degree from Yale and had spent a year in Europe. The University of Michigan was offering the doctor's degree and had Cole and Ziwet on its staff. Ziwet also was trained in Europe. Most of the colleges were offering mathematics as far as the calculus; those that were going further depended largely on men who had been trained in Europe. The American Journal of Mathematics, established by Johns Hopkins in 1878 with Sylvester GEORGE ABRAM MILLER 259 as editor, depended for the first few years on contributions by Euro- peans. The Annals of Mathematics (1884) was edited by Ormond Stone at the University of Virginia. The Bulletin of the New Yor\ Mathematical Society (1891) was edited by Fiske, Jacoby, and Ziwet. The New York Mathematical Society had 23 members in 1890; Miller was elected to membership in 1891. The Society changed its name to the American Mathematical Society in 1894. Except in a few institutions, mathematics libraries were non-existent; for ex- ample, the mathematics books at the University of Illinois occupied one fifteen-foot shelf in 1893. Miller had no considerable contact with mathematicians or with their works before he went as Instructor to the University of Michi- gan in 1893, and he had behind him no family tradition of erudition or scholarship to spur him on. He began teaching school at seventeen to earn money to permit him to continue his education. He attended Franklin and Marshall Academy, a subdivision of the College at Lancaster, in 1882-1883. He attended Muhlenburg College from January, 1884, to 1887, receiving the degree of Bachelor of Arts with honorable mention, and ranked third in his class of twelve. Muhlenburg granted him the degree of Master of Arts in 1890. The catalogue of Muhlenburg states that the recipient of the degree should be of good moral character, should have been a Bachelor of Arts for three years, and should have been engaged in liberal and professional pursuits. During the year 1887-1888 Miller was Principal in the schools of Greeley, Kansas. From 1888 to 1893 ne was Professor of Mathematics at Eureka College in Eureka, Illinois. In 1892 he received the degree of Doctor of Philosophy from Cum- berland University in Lebanon, Tennessee. It is not known that he ever went to Lebanon, although it is likely that he wrote some examinations there. He was registered as a graduate student at Cum- berland during the year 1891-1892, but graduate work could be taken by correspondence and Miller was teaching at Eureka during the year. A thesis was a requirement for the degree, but examinations in the advanced courses could be substituted for the thesis. He was 260 BIOGRAPHICAL MEMOIRS offering the same courses and the doctor's degree at Eureka and was using the same texts. He had a graduate student taking these courses for two years but the student did not receive the doctor's degree, although he did succeed Miller as Professor of Mathematics at Eureka. In the summers of 1889 and 1890 he went to Johns Hopkins and the University of Michigan, although neither univer- sity was in session during these summers. If he had contact with the mathematicians at these places there is no trace of it to be found. When he went to the University of Michigan as Instructor in 1893 he lived in Cole's home, and he credited Cole with starting him on his career in the theory of groups. But if he met Cole in the summer of 1890 no spark was struck, for if there had been Eureka would have had a course in the theory of groups. He spent the years 1893-1895 as Instructor at Michigan. During the years 1895-1897 he was in Europe attending the lectures of Sophus Lie at Leipzig and Camille Jordan in Paris, and wrote prodigiously on the theory of groups. He is, therefore, numbered among the group of mathematicians with European training. He credited Lie with starting him on a systematic study of commutators and commutator subgroups, but otherwise Lie's influence on him was small. Neither Lie nor Miller was greatly interested in the other's groups. Jordan was interested in questions of primitivity and imprimitivity and Miller kept returning to these questions for the remainder of his life. It is doubtful, however, that Jordan influ- enced him greatly, for Miller worked on his own problems with his own methods and he never engaged in the explanation of Jordan's works or Jordan's methods. From 1897 to I9°I Miller was Assistant Professor at Cornell; during 1901 to 1906 he was Assistant Professor and Associate Professor at Leland Stanford; from 1906 he was at the University of Illinois serving as Associate Professor, Professor, and from 1931 Professor Emeritus. He was married in 1909 to Cassandra Boggs of Urbana, Illinois. They had no children and Mrs. Miller died in 1949. The catalogues of Eureka College for the years 1887 to 1895 GEORGE ABRAM MILLER 261 reveal the G. A. Miller that was known to the writer by daily asso- ciation during his last thirty years. Under the heading of the Depart- ment of Mathematics in the Eureka catalogue of 1889-1890 appears the following: ". . . The study of mathematics— a. Develops the power of reasoning. b. Cultivates precise methods of reasoning. c. Trains the mind in abstract thinking. d. Lifts from servile imitation to original thinking. e. Gives a steady and dignified bearing to the mind. f. Enables the student to understand all the other sciences better. g. Cultivates the habit of persistent and well-directed exertion, and h. Cultivates the habit of distinguishing clearly between the known and the unknown. Authority for all these statements may be found in the works of the greatest philosophers. . . ." One can see the young, meagerly trained Professor of Mathe- matics called upon by the President for copy for the catalogue; he is not content to offer the customary list of requirements and lauda- tory but vague assertions about the value of mathematics; he thinks out and states in detail his philosophy of education and the part that mathematics has in it. Sixty years later he would have been more precise in referring to the philosophers, but the goals he set up in 1889 were the ones toward which his whole life was directed. The statements under Mathematics in the catalogues change from year to year. In 1890 Miller offered the doctor's degree. New ad- vanced courses were offered. The description of one of these in 1892-1893 reads: "The Higher Algebra and Determinants have been prepared expressly for the students of this institution. The treatments of the following subjects are more comprehensive than those given by any other American text-book on Higher Algebra, Logarithms, Choice and Chance, Apparent Paradoxes, Determinants, and Theory of Numbers. The other subjects commonly treated in works of 262 BIOGRAPHICAL MEMOIRS this kind are introduced and discussed according to the most ap- proved methods." The book to which this refers was published under the title Determinants in 1892 in the Van Nostrand Science Series. It could have been said of him in 1893 that here is a man of power who appreciates his own worth, who claims his place in the sun, and who, while vigorous and astute in seeking that place, offers solid achievement as the basis of the claim. The publication of lists of all the substitution groups of low de- grees was started in 1850 when a French mathematician, Serret, listed all the 19 possible groups of degrees not greater than five. Miller came upon the scene in 1893 to complete the lists for degrees eight and nine. The principles underlying the determination of all the groups of a given degree were then fairly clear. Miller put them to use and rapidly went on to determine the 994 intransitive groups of degree ten; these brought to 1,039 tne total number of groups, transitive and intransitive, of degree ten. Jordan had determined the eight transitive groups of degree eleven in 1872. Miller and a student, a G. H. Ling, determined the 1,492 intransitive groups of degree eleven in 1901. Miller did the transitive groups of degree twelve in 1896; the transitive groups of degrees thirteen and four- teen and the primitive groups of degree fifteen in 1897; the primi- tive groups of degree sixteen in 1898; and the transitive groups of degree seventeen in 1899. The subject of the determination of sub- stitution groups, which had interested many mathematicians besides Cayley and Jordan, was virtually taken over by Miller in 1893, and he finished it in the sense that nobody has tried to go beyond him and probably nobody has followed him so far. His work on sub- stitution groups could be carried on but probably will not, unless someone should see how to put a high-speed computer to work on it. Any obvious attack with a computer seems sure to lead directly to insuperable difficulties. Miller's belief in persistent and well-directed exertion would have taken him beyond the transitive groups of degree seventeen if he had not early become interested in abstract groups. Every substitu- GEORGE ABRAM MILLER 263 tion group is a representation of one and only one abstract group; an abstract group may have many representations as a substitution group on a given number of letters. In determining all the substitu- tion groups of a given degree, it is necessary to know much about the abstract groups that can be represented on smaller numbers of letters. Every abstract group can be represented in one and only one way as a regular substitution group, that is, a group which is tran- sitive and has its degree equal to its order. Miller's first work on abstract groups was done in terms of regular substitution groups, and throughout his life he returned to regular substitution groups to study certain properties of abstract groups. There is an element of the fantastic in his first publication on abstract groups. In 1884 Felix Klein used a certain method to prove that a group of order 60 was simple. In a paper dated November 5, 1894, E. H. Moore used the same method to prove that a group of order 168 was simple, and asked if anybody knew of a simple group on which that method would not work. On December 28 Miller gave him an example, the alternating group on 68 letters. The non-mathematical reader may be interested in knowing that the alternating group on 68 letters has order one half of 68!, that 68! is the product of all the integers from 1 to 68, and that this is a number which if written out would require about two lines on this page. One man displays an inclina- tion to generalize from his experience with the two smallest simple groups, and the other, though a novice, answers the question even if he has to go beyond astronomical figures to do it. Miller's first considerable contribution to abstract groups, also done in terms of regular substitution groups, was his list in 1896 of all the abstract groups of orders less than 48. This paper con- tained the first determination of the 15 groups of order 24 and the 51 groups of order 32. The enumeration of all the abstract groups of low order had been begun by Cayley in 1854 when he proved that there are just two groups of each of the orders 4 and 6; in 1859 Cayley determined the 5 groups of order 8, and in 1889 the 5 groups of order 12. The 14 groups of order 16 were determined 264 BIOGRAPHICAL MEMOIRS in 1893 by two men independently, J. W. A. Young, an American student under Bolza at Clark, and Holder in Germany. Shortly before Miller's paper in 1896 a French mathematician, Le Vavasseur, had announced in the Comptes Rendus that he had found 75 groups of order 32 and that he had not yet reached the end. Two years later an Italian, Bagnera, stated in the Annali di Matematiche that Miller had made a mistake and there were 50 groups of order 32, but the following year he agreed that the number is 51. The orders for which there are the largest numbers of distinct abstract groups are orders which are divisible by a high power of a prime number; the smallest of these are thus the powers of 2: 2, 4, 8, 16, 32, 64, 128. In 1930 Miller determined the 294 groups of order 64; he estimated at that time that there are more than a thousand groups of order 128. He had determined the 52 groups of order 48 and the 15 groups of order 54 in 1898; he did those of order 54 by showing that there are just 15 groups of order 2ps for every odd prime p. He determined the 57 groups of order 168 in 1902, the groups of order 72 in 1929, and those of order 96 in 1930. In the 34 years between the publication of his lists of the groups of orders 32 and 64 only one man, Potron, had offered a list of the groups of order 64 and that was not a suc- cess. Miller thus took over the enumeration of abstract groups at order 24 and carried it to order 100. The enumeration has since been carried to order 160, omitting order 128. The two most fundamental questions about groups are: What groups exist? How can one group be distinguished from another? Information sufficient to determine a particular group will be enough to determine all of its subgroups, and sufficient information about subgroups and their relations will be enough to determine the group. Thus these lists of substitution groups of low degrees and of abstract groups of low orders not only answer the first question as far as they go, but they provide tools for the investigation of the groups beyond. The second question above may be restated: What questions should one ask of a group so that the answers will enable one to