The structure of the lattice LB of varieties of idempotent semigroups or bands (as universal algebras) was determined by Birjukov, Fennemore and Gerhard. Wis- math determined the structure of a related lattice: the lattice LBM of varieties of band monoids. In the first two parts we study several questions about these varieties. In Part I we compute the cardinalities of the Green classes of the free objects in each variety of LB [LBM]. These cardinalities constitute a useful piece of information in the study of several questions about these varieties and some of the conclusions obtained here are used in parts II and III. Part II concerns expansions of bands [band monoids]. More precisely, we compute here the cut-down to generators of the Rhodes expansions of the free objects in the varieties of LB. We define Rhodes expansion of a monoid, its cut-down to generators and we compute the cut-down to generators of the Rhodes expansions of the free objects in the varieties of LBM. In Part III we deal with Eilenberg varieties of band monoids. The last chapter is particularly concerned with the description of the varieties of languages corresponding to these varieties.

It has been found elsewhere that when approximate relations for the two-particle correlation functions of classical statistical mechanics, such as the Percus-Yevick and the mean-spherical approximations, are applied to the lattice gas models with nearest-neighbour interactions simple expressions are obtained for the total correlation function in terms of the lattice Green's function. Since many of the properties of random walks on a lattice can be described by the lattice Green's function, it follows that these systems, at least when treated under these approximations, may be analysed in terms of the language of random walks. Here the theory of random walks on lattices is appropriately extended to show that the relationship between the correlation functions and the lattice Green's function is not dependent upon the employment of these approximations, nor to a particular range or form of the potential function. Instead, this relationship is shown to be an alternative form of the Ornstein-Zernike relation between the direct and total correlation functions. The direct correlation function is directly related to the probability of a single step, whereas the total correlation function is given by the first-passage- time probabilities of the random walks. Thermodynamic properties, such as the isothermal compressibility, are also interpreted in terms of random-walk concepts. The random-walk formulation is then extended to include the study of ordered phases in lattice-gas models and hence order-disorder transitions in these systems. Also, an asymptotic form for the lattice Green's function is derived to illustrate how the form of decay of the total correlation function at large distances depends on the properties of the potential function. To show that the random-walk interpretation of the Ornstein-Zernike relation is not restricted to lattice systems, we define analogous random-walk functions for continuous space. These lead to a straight-forward generalization of most expressions for discrete space-; the relationship between the continuous-space total correlation and Green's functions has the same form as that for the lattice systems. We also explore the possibility of obtaining random-walk properties of a (lattice or continuous-space) system, not from the existing approximate expressions for the direct correlation function, but instead from a generalised Ornstein-Zernike relation based on a maximum principle of classical statistical mechanics. Finally, we choose a few specific lattice-gas models to show how the method describes their different properties, such as the behaviour of the total correlation function or that of an order- disorder phase transition.

M.Sc. (Mathematics) / Please refer to full text to view abstract

In this dissertation we completely determine the maximal subgroups of the following finite simple groups: (i) POgX?) and 3D^q) for all prime powers q (ii) 2G2(32m+1) for all integers m (iii) G2(<7) for all odd prime powers q. Moreover, if Go is one of the groups appearing in (i), (ii) or (iii), then we also determine the maximal subgroups of all groups G satisfying: GO<G< Aut{Go\ (*) where Aut{Go) is the automorphism group of Go. Chapter 1 is devoted to the case Go = PClt(.q), where q = pt and p is prime. We first analyse the structure of the full automorphism group A = Aut(Go), as follows. Let Q be a quadratic form of Witt defect O defined on an 8-dimensional vector space V over F = GF(q). We write 0 = 0 (V,F£) for the isometry group of Q. We then define a chain of groups 0 <. SO < O < A < T all related to the geometry (V,¥,Q). The group T is the full semilinear group associated with Q and fl = [0,0] is a perfect group. Upon factoring out scalars, we obtain the projective groups PCI < PSO < PO < PA < PI\ We have Ptl = Go and | A:PT \ = 3. In fact, A is generated by Pr and a triality automorphism, which occurs because the Dynkin diagram of Go admits a symmetry of order 3. We then show that AlGo — Ex Z/, where E is the symmetric group S3 or S4. We thus obtain a homomorphism JT : A —» E whose kernel is isomorphic to GoXf. It turns out that G (as in (*)) contains a triality automorphism if and only if 3 divides | r(G)\. A recent theorem of M. Aschbacher [Invent, meth. 76 (1984), 469-514] shows that if G < PV, then the maximal subgroups of G fall into two families, which we may call C and S. Groups in C can be read off from from Aschbacher's paper, and we determine the groups in S by studying the p- modular representations of the finite simple groups. Thus we appeal to the classification of the finite simple groups. We then consider the case in which G •%. PY. Here G contains a triality automorphism and our argument goes roughly like this. Take Af to be a maximal subgroup of G which satisfies MGO = G and write M o = M n Go. Then M o < L < Go for some maximal subgroup L of Go. But M contains a triality automorphism T and so M o < L n U n Lr2. Now L is known because we have already handled the case in which G < PT (in particular, the case G = Go). Therefore our knowledge of L together with our knowledge concerning the action of r allows us to determine all possibilities for Mo. Hence M is known, for M £- MO.(G/GO). In Chapter 2 we treat the case Go = aD^(q). The group 3D4(<7) is the centralizer in PO^O?3) of a suitable triality automorphism. Thus the information about triauty which we collect in Chapter 1 is exploited in Chapter 2 to obtain the maximal subgroups of 3D^(q) and it automorphism groups. Similarly, G2O7) is the centralizer in PCl^iq) of a suitable triality. Thus in Chapter 3 we deal with the case Go = G2(?) (with q odd) by exploiting triality once again. Our methods for analysing G2O7) readily lend themselves to handle Go = 2Gi{q\ and this work is presented in Chapter 4. Chapter 4 also contains information about the maximal subgroups of the automorphism groups of the Suzuki groups Sz(q) = ^i^fa)- Note that in his original paper, Suzuki find the subgroups of the simple group We however find the maximal subgroups of all groups G satisfying < G < Aut(Sz(q)). In Chapter 5 we present lists of maximal subgroups of several families of low dimensional finite classical groups, including PSLn(q) for 2 < n < 11. We do not include proofs, although we sketch a proof for PSL&(q). Some of these results have appeared much earlier in the literature (dating as far back as the 19th century), but most of them are new.

Magnetic space groups (MSGs) were first introduced (under a different name) by Heesch more than 30 years ago, and a list of all of them was published by Belov, Neronova and Srairnova in 1955. However, no mathematically rigorous derivation of MSGs can be found in the existing literature, although an outline of a method for obtaining a large class of MSGs was published by Zaraorzaev in 1957. In this thesis a systematic rigorous method for constructing MSGs is described in detail, and a proof that the method in fact gives all the MSGs is presented. The method also leads in a natural way to a classification of MSGs which is useful for a systematic study of the arrangements of spins in ferromagnetic, ferrimagnetic and antiferromagnetic crystals. The first and the last chapter of the thesis deal with the physical aspects of the problem, the remaining chapters with purely mathematical aspects of it. / Science, Faculty of / Physics and Astronomy, Department of / Graduate

The study of invariant means on spaces of functions associated with a group or semigroup has been the interest of many mathematicians since von Neumann's work on invariant measures appeared in 1929. In recent years, many important properties of locally compact groups have been found to depend on the existence of an invariant mean on a suitable translation-invariant space of functions on the group. In this thesis, we deal mostly with invariant means on the space L∞G) of bounded measurable functions on a locally compact group G. Several characterisations of the existence of an invariant mean on L∞G) are given. Among other results, we prove the remarkable theorem that L∞(G) has a left invariant mean if and only if G is topologically right stationary, an analogue of a recent result for semigroups by T. Mitchell. However our approach is entirely different. / Science, Faculty of / Mathematics, Department of / Graduate

The purpose of this paper is to display in some detail the theory of groups with operators leading to the Jordan-Holder Theorem on composition series. A number of examples are given to illustrate the theory. The author hopes that the material which follows is more easily read than the well known sources on group theory. In 1832, Evariste Galois introduced the concept of group in a letter written to a friend on the eve of his tragic death in a duel at the age of twenty-one. The Norwegian mathematician Neils Hendrick Abel, among others, has contributed much to the development of group theory; Wolfgang Krull and Emmy Noether in particular made notable contributions to the theory of groups with operators. The theory of groups has found widespread application in many branches of mathematics. / Advisor: Nickolas Heerema, Professor Directing Paper. / Typescript. / "January, 1954." / "Submitted to the Graduate Council of Florida State University in partial fulfillment of the requirements for the degree of Master of Science." / Includes bibliographical references (leaf 44).

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