AUTOBIOGRAPHY. I Was born in Chicago on the twenty-fourth day of October, 1879 After completing the course in the primary and secondary schools of the City of Chicago, I entered the College of Liberal Arts of Northwestern University in September, 1896. I was graduated in June, 1900, receiving the degree Bachelor of Arts. The next two years I spent in graduate work in Mathematics and Physics at Harvard University, and received the degree Master of Arts in June, 1902. I continued my work for two years in the Graduate School of the University of Chicago as a Fellow in Mathematics. I wish to acknowledge my indebtedness to all of the instructors under whose direction I have worked as a graduate student, and to express my appreciation of the kindly interest manifested by Professor Dickson, under whose direction this investigation has been carried out. WILLIAM HENRY BUSSEY. SIMPLY ISOMORPHIC WITH THE GROUP LF[2,pn] By W. H. Busset. [Received December 10th, 1904.—Read January 12th, 1905.] [Extracted from the Proceedings of the London Mathematical Society, Ser. 2, Vol. 3, Part 4.] Introduction. 1. The object of this paper is the proof of the following six theorems concerning sets of generational relations for the abstract group G, simply isomorphic with the group LF [2, pn~] of all linear fractional transformations, on one variable, having determinant unity and coefficients belonging to the GF(pn). Theorem I.*—The abstract group GjP(p>_i), simply isomorphic with the group LF [2, p], p > 2, may be generated by two operators T and S, subject to the generational relations (A) &> = I, T2 = /, (ST)8 = I, (SrTS2TT)2 = I, T 0. Theorem II.—The abstract group G^^^, simply isomorphic with the group LF[2, p*], p > 2, n > 1, may be generated by (pn-f-l) operators T and Sx, X running through the marks of the GF(pn), subject to the generational relations (1) S0 = I, SxSp = S*+Il (X, n any marks), (2) T*=I, (S1Tf = I, (J3) (8) (S» TSs/r Tf = I (t any mark =f= 0), ,(4) [1/a, a2], [1/a, ia4], [t, a], a] (a =£ 0), where i is a primitive root of the GF(pn), and a is any mark subject to a restriction implied in the notation [X, ft]. * For the special cases in which pn < 47, this theorem has been proved by Prof. Dickson, Proc. London Math. Soc., Vol. xxxv., pp. 292-305; Bull. Amer. Math. Soc, Vol. ix., p. 297. Note.—The symbol [X, M] is used to denote the relation* where X, n are marks such that X/x 1. Theorem III. t—For the special cases in which pn = 49, 81, 121 relations (1), (2), (8) form a set of generational relations for G^^-v), P > 2. Theorem IV. t—The abstract group G2,.(2-2»-i), simply isomorphic with the group LF[2, 2n], may be generated by three operators a, b, c, subject to the generational relations (5) a2n-1 = I, 62 = J, ba(b = a"6af, (C) | (6) c* = I, (ca)a = /, (cb)3 = I, where £ = 1, 2, 8, (2n—2), and i?, £ are determined by the relations i( — l+i(, >] = £—£, mod (2n—1), i being a primitive root of the GF(2n). Theorem V.—The abstract group G2n(2s»_i), simply isomorphic with the group LF{2, 2n], may be generated by two operators a and d subject to the generational relations CD) a2n^ -I, d* = I, (da(d^atf = I, (datda()e = I, where £ = 1, 2, 8 (2n—2), and £ is determined by the relation is = l+if, i being a primitive root of the GF(2n). Theorem VI.§—In the special cases in which n = 2, 8, 4, 5, 6, the abstract group G2»(2jb_1), simply isomorphic with the group LF [2, 2n], * Relations (1), T* = I, [A, pi], A, n any marks suoh that A^=j£ 1, constitute a set of generational relations for 0. This is a special case of a more general theorem valid for any field due to Moore. See Proc. London Math. Soc, Vol. xxxv., p. 293, and Dickson's Linear Groups, p. 300. Note that, when \ - 0 or 1, [A, n] reduces to (5, T)% = /, and, when A = — 1, [A, p] reduces to (3). t For the special cases in which pn = 9, 25, 2", 125, 243, Prof. Dickson has proved that (1), (2), (3) constitute a set of generational relations for ^jj,'>(p!'>-i), P> ^> ^e proof" for the cases in which pn = 125, 243 have not been published. X This theorem is due to de Seguier, Journal de Mathcmatiques, Tome vm., p. 253. § The set of generational relations (E) is due to Prof. Dickson. He has proved Theorem VI. for n = 2, 3, 4. See Proc. London Math. Soc, Vol. xxxv., p. 306 and p. 443; Bull. Amur. Math. Soc., Vol. ix., pp. 194-204. For n = 2, the set (E) reduces to Ah = /, B1 = /, (ABf = J. may be generated by two operators A and B subject to the generational relations (E) Ar+1 = I, B2 = I, (AB)a = I, (BArBAf = I, where r = 1, 2, 8, 2n, and the value of s is determined by the relation f(f+ts+l) = P(**+1)+1, toeing defined by the relation f = t*f+l, i being a primitive root of the GF(1n). The Group G Simply Isomorphic With LF [2, pn\ p > 2. 2. Lemma.—The abstract group G^pK(Jp>_Y), simply isomorphic with the group LF[2, pn], p > 2, may be generated by (pn+2) operators R, T, and Sx, X running through the marks of GF(pn), subject to relations (1) and (7) fl*"n^) = I, (8) SxR' = R'Stf* (X any mark, a- = 0 or any integer), (9) (TR')* = I (a- = 0 or any integer), (10) TSyT = R"S-yTS-yy (y any mark =£0, if = —y), i being a primitive root of the GF(pn). Proof. — The group LF[2, pn~\, p>%, may be generated by the pn+l transformations T :V = —, Sx : z' = z+\ (X any mark), z while the sub-group K of the transformations may be generated by the transformations S1.X = SA : J — z+\, B:z' = izli-\ i being a primitive root of the GF(pn). These generators of K satisfy relations (1), (7), and (8). Since the group LF [2, p"\ p > 2, when represented as a permutation group on (pn-\-l) letters, is doubly transitive while the sub-group K, being then a permutation group on pn letters, is simply transitive, it follows from the work of Jordan* that it is possible to determine y, S, 17, f, and u such that "Traite des Substitution*, p. 32. |