Topics in Group Theory

The Frattini argument shows that, if N is a normal subgroup of a group G and if S is an intravariant subgroup of N, then G=~(S)N. Unfortunately, there appears to be no completely satisfactory Lie algebra analogue of the concept of an intravariant subgroup. We get around this difficulty by considering ways in which intravariant subgroups arise, and investigating the analogous situations for Lie algebras. In finite group theory, the best known examples of intravariant subgroups are 3s for suitable classes 3s In particular, Sylow p-subgroups are the projectors for the class of finite p-groups, a class which is minimal subject to the existence of projectors in all finite soluble groups. We show, subject to suitable conditions on N and 3s that if N is an ideal of the Lie algebra L and S is an 3s of N, then L=N+~(S). In particular, this holds if S is a Cartan subalgebra of N. All Lie algebras considered are finite-dimensional over some field F. We write U__< L, U~ L for U is a subalgebra, respectively ideal, of L. We denote the centraliser and normaliser of U in L by cgL(U ) andS(U) respectively./2 is the derived algebra and ~b(L) is the Frattini subalgebra of L. If M is an L-module, then ML={rn~M[mx=O for allx~L}.We denote by 9t and ~ the classes of all nilpotent and all soluble Lie algebras respectively. Let 3s be a homomorph of Lie algebras. The subalgebra S of L is called an 3s of L if

We develop the theory of semisimplifications of tensor categories defined by Barrett and Westbury. In particular, we compute the semisimplification of the category of representations of a finite group in characteristic p in terms of representations of the normalizer of its Sylow p-subgroup. This allows us to compute the semisimplification of the representation category of the symmetric group Sn+p in characteristic p, where 0 ≤ n ≤ p − 1, and of the Deligne category $$ \underline { \mathop {\mathrm {Rep}} \nolimits }^{\mathrm {ab}}S_t$$ , where t ∈ℕ. We also compute the semisimplification of the category of representations of the Kac-De Concini quantum group of the Borel subalgebra of $$\mathfrak {sl}_2$$ . We also study tensor functors between Verlinde categories of semisimple algebraic groups arising from the semisimplification construction and objects of finite type in categories of modular representations of finite groups (i.e., objects generating a fusion category in the semisimplification). Finally, we determine the semisimplifications of the tilting categories of GL(n), SL(n), and PGL(n) in characteristic 2. In the appendix, we classify categorifications of the Grothendieck ring of representations of SO(3) and its truncations.

Weights for symmetric and general linear groups

The representation theory of finite groups began with the pioneering research of Frobenius, Burnside, and Schur at the turn of the century. Their work was inspired in part by two largely unrelated developments which occurred earlier in the nineteenth century. The first was the awareness of characters of finite abelian groups and their application by some of the great nineteenth-century number theorists. The second was the emergence of the structure theory of finite groups, beginning with Galois’ brief outline of the main ideas in the famous letter written on the eve of his death and continuing with the work of Sylow and others, including Frobenius himself.

The group SL(3,Z) cannot act (faithfully) on the circle (by homeomorphisms). We will see that many other arithmetic groups also cannot act on the circle. The discussion will involve several important topics in group theory, such as ordered groups, amenability, bounded generation, and bounded cohomology. Lecture 1 provides an introduction to the subject, and uses the theory of left-orderable groups to prove that SL(3,Z) does not act on the circle. Lecture 2 discusses bounded generation, and proves that groups of the form SL(2,Z[a]) do not act on the real line. Lectures 3 and 4 are brief introductions to amenable groups and bounded cohomology, respectively. They also explain how these ideas can be used to prove that actions on the circle have finite orbits. An appendix provides hints or references for all of the exercises. These notes are slightly expanded from talks given at the Park City Mathematics Institute's Graduate Summer School in July 2012.

On the computational complexity of the theory of Abelian groups

American Journal of Mathematics, vol. xviii. No. 2, April.—The intermediary orbit, i.e. the Moon's periodic orbit relatively to the Sun obtained from the variation terms when all terms but those depending on the ratio of the mean motions only are omitted, has been considered in vol. i, by Dr. Hill, and subsequently in the Acta Manthematica (vol. viii.) the same writer obtained an expression for the motion of the Moon's perigee, so far as it depends on the ratio of the mean motions. These papers have been followed by others by Prof. E. W. Brown, in which the terms depending on the solar parallax and the lunar eccentricity are computed.—The object of the opening paper of the present number, on the inclinational terms in the Moon's coordinates, by P. H. Cowell, is to take into account, according to Dr. Hill's method, the inclination of the orbit, considering it as being the manifestation of a small oscillation about Dr. Hill's distorted circular orbit, which relatively to the Sun is a closed curve. The terms multiplied by the first power of the inclination have been calculated to the sixth order, and an expression for the part of the motion of the Moon's node, that depends upon the mean motions only, has been found as far as the eighth order, i.e. one term further than in Delaunay's series. The terms multiplied by the square of the inclination have been calculated to the fifth order, and the terms multiplied by the third power of the inclination to the fourth order in m. The notation adopted is that of the paper by Prof. Brown (Am. Journ. Math., vol. xvii.).—A short note by A. S. Chessin, on non-uniform convergence of infinite series, brings out more clearly a point in a previous note (vol. xviii. No. 1), which the writer says has been misunderstood.—On a certain class of equipotential surfaces, by B. O. Peirce, discusses the nature of such systems of plane curves as are at once the right sections of possible systems of equipotential cylindrical surfaces belonging to distributions of matter which attract, according to the law of nature, and the generating curves of possible systems of equipotential surfaces of revolution.—M. Petrovitch contributes “Remarques sur les équations de dynamique et sur le mouvement tautochrone.”—A note on C. S. Peirce's paper on a quincunclal projection of the sphere, by J. Pierpont, corrects an inaccuracy in that paper (vol. ii. p. 394). Mr. Pierpont, in a note on the invariance of the factors of composition of a substitution-group, gives a much simplified proof of this important theorem.—H. Maschke, in a long article (pp. 156-188) on the representation of finite groups, especially of the rotation—groups of the regular bodies of three-and four-dimensional space, by Cayley's colour diagrams, shows that Cayley's method (the theory of groups, graphical representation, Am. Journ., vol. i., and on the theory of groups, Am. Journ., vol xi.) can be readily applied to the construction and investigation of numerous groups of higher orders. In particular, the writer says, the colour diagrams for the rotation groups of the regular bodies can be arranged in such a way that they lend themselves much easier, at least in some respects, to a study of the groups concerned, than even the models of the regular bodies. Numerous diagrams of interest accompany the paper.

The main problem in the theory of groups is to classify them, and to study their structures. The present chapter deals with the Sylow theorems which play a very crucial role in the structure theory of groups. This chapter also contains the classification of finite abelian groups together with the Schreier and the Jordon-Holder theorems.

Finite Groups With Many Conjugate Elements

In this survey, we discuss some basic problems in representation theory of finite groups, including some long-standing conjectures of Alperin, Brauer, and others. A possible approach to some of these problems is to use the classification of finite simple groups to reduce the problem in consideration to some, more specific, questions about simple groups. We will describe recent progress on reduction theorems in this direction. We will also outline applications of these results to various problems in group theory, number theory, and algebraic geometry.

A computational method for determining the automorphism group of a finite solvable group

A subgroup H of a finite group G is called c-normal in G if there exists a normal subgroup N of G such that G = HN and $H \cap N \leq H_{G} = {\rm core}_{G}(H)$. In this paper, we investigate the class of groups of which every maximal subgroup of its Sylow p-subgroup is c-normal and the class of groups of which some minimal subgroups of its Sylow p-subgroup is c-normal for some prime number p. Some interesting results are obtained and consequently, many known results related to p-nilpotent groups and p-supersolvable groups are generalized.

Suppose G is a finite group and p and q are distinct odd primes. Let χ \chi be an irreducible character of G and x and y be a p-element and a q-element of G such that χ ( x ) , χ ( y ) \chi (x),\chi (y) are both irrational. In this situation it is known that G contains an element of order pq. John Thompson has asked whether y must commute with some conjugate of x. We show by example that this need not be the case.

Glauberman’s Z *-theorem [1] and the theorem of Bender are two most important tools for local analysis in the theory of finite groups. The Z *-theorem generalizes the known Burnside and Brauer–Suzuki theorems on finite groups with cyclic and quaternion Sylow 2-subgroups. Whether these theorems are valid in a class of periodic groups is unknown. We prove that the Z *-theorem is invalid in the class of all periodic groups. In particular, this gives negative answers to questions of A. V. Borovik [3, Question 11.13] and V. D. Mazurov [3, Question 17.71a].

: A unified approach to the design and evaluation of fast algorithms for discrete signal processing is developed. Based on the theory of finite groups, it hence includes the familiar cases of the fast Fourier and Walsh- Hadamard transforms. However, the use of noncommutative groups reveals a large variety of novel methods. Some of these exhibit a superior performance, as measured by both reduced error rates and computational complexity, on nonstationary data. The recent history of this subject is reviewed first, followed by a detailed examination of the three principal ingredients of the present study: the underlying groups, the signal-processing tasks on which the group-based algorithms are to compete, and the signal models used to define the data environment. Test results and conclusions then follow, the former being based on the use of random correlation matrices. (kr)