Thesis (PH. D.)--University of Michigan, 1943. / Thesis note on label mounted on cover. "Presented to the society, November 27, 1943." Reprinted from the Transactions of the American mathematical society, vol. 56, no. 3 ... November, 1944. Bibliography: p. 536.

Inaug.-diss.--Albertus-Universität.

By a linear group we shall mean essentially a group of invertible matrices over a ring. Thus, we include in our class of linear groups the 'classical' geometric groups. These are the general linear group, GL[sub]n(F), the orthogonal groups, 0[sub]n (F) and the syraplectic groups Sp[sub]n(F). The normal and subnormal subgroup structure of these groups is well known and has been the subject of much investigation since the turn of the century. We study here the normal and subnormal structure of some of their infinite dimensional counterparts, namely, the infinite dimensional linear group GL(Ω,R), for arbitrary rings R, and the infinite dimensional syraplectic group Sp(Ω,R), for commutative rings R with identity. We shall see that a key role in the classification of the normal and subnormal subgroups of GL(Ω,R) and Sp(Ω,R) is played by the 'elementary' normal subgroups E(Ω,R) and ESp(Ω,R). We shall also see that, in the case of the infinite dimensional linear group, the normal subgroup structure depends very much upon the way in which R is generated as a right R-module. We shall also give a presentation for the 'elementary' subgroup E(Ω,R) when R is a division ring.

If d(M) denotes the rank of the Schur multiplicator of a finite group G, then a group is efficient if -def G = d(M). Efficient presentations of the simple groups PSL(2,p), p an odd prime > 3, were obtained by J.G. Sunday. This raised the question of whether or not all finite simple groups are efficient. In this thesis, we investigate the deficiency of the groups PSL(2,pn). J.A. Todd gave presentations for PSL(2,pn) which use large numbers of generators and relations. Starting with these, we obtain, at best, deficiency -1 presentations for PSL(2,2n) (= SL(2,2n)) and deficiency -6 presentations for PSL(2,pn), p an odd prime. If pn = -1(mod 4), the latter can be reduced to a deficiency -4 presentation. Efficient presentations for PSL(2,25), PSL(2,27) and PSL(2,49) are obtained. The Behr-Mennicke presentation for PSL(2,p) is one of the most fundamental in the sense that it forms the basis for others, such as those given by Sunday, Zassenhaus and Sidki. Behr and Mennicke derived their presentation indirectly, and it would be desirable to have a more direct proof. The groups G[sub]p(a) are defined as < U, R, S | U3 = (UR)2 = (US)2 = Sp = Rt = (SaRU)3= 1, Sa2R = RS > where a ε GF(p)* and a2t = 1 (mod p) . We show that G[sub]p (2) is isomorphic with the Behr-Mennicke presentation for PSL(2,p), p > 3. Conditions are found to discover when Gp (a) is isomorphic with PSL(2,p) and, under these conditions, this provides a direct proof of the Behr-Mennicke presentations. For any odd positive integer m, we show that the groups SL(2,Z (m)) and PSL(2,Z(m)) are efficient.

No description available.

Akademische Abhandlung--Helsinki. / Includes bibliographical references (p. [38]).

No description available.

Inaug.-diss.--Zürich.

No description available.

No description available.