On Isomorphism Testing of Groups with Normal Hall Subgroups

A normal Hall subgroup N of a group G is a normal subgroup with its order coprime with its index. Schur-Zassenhaus theorem states that every normal Hall subgroup has a complement subgroup, that is a set of coset representatives H which also forms a subgroup of G. In this paper, we present a framework to test isomorphism of groups with at least one normal Hall subgroup, when groups are given as multiplication tables. To establish the framework, we first observe that a proof of Schur-Zassenhaus theorem is constructive, and formulate a necessary and sufficient condition for testing isomorphism in terms of the associated actions of the semidirect products, and isomorphisms of the normal parts and complement parts. We then focus on the case when the normal subgroup is abelian. Utilizing basic facts of representation theory of finite groups and a technique by Le Gall (STACS 2009), we first get an efficient isomorphism testing algorithm when the complement has bounded number of generators. For the case when the complement subgroup is elementary abelian, which does not necessarily have bounded number of generators, we obtain a polynomial time isomorphism testing algorithm by reducing to generalized code isomorphism problem, which asks whether two linear subspaces are the same up to permutation of coordinates. A solution to the latter can be obtained by a mild extension of the singly exponential (in the number of coordinates) time algorithm for code isomorphism problem developed recently by Babai et al. (SODA 2011). Enroute to obtaining the above reduction, we study the following computational problem in representation theory of finite groups: given two representations ρ and τ of a group H over \( \mathbb{Z}_p^d \), p a prime, determine if there exists an automorphism : H → H, such that the induced representation ρ𝜙 = ρ ◦ 𝜙 and τ are equivalent, in time poly(|H|, p d).

Modular representations of finite groups and Lie theory

This is a survey on some recent results in representation theory of finite groups of Lie type which were derived in joint work with J. Du. We begin with two generalizations of Green’s theory of vertices and sources: First by allowing as vertex of an indecomposable representation of a finite group G not only subgroups of G but also subfactors. In addition we derive a relative version of the notion of a vertex (and a source) by considering socalled Mackey systems of subfactors. As a consequence one gets for example the Harish-Chandra theory for irreducible ordinary representations of finite groups of Lie type as special case. Secondly we extend the classical theory of vertices and sources to Hecke algebras associated with Coxeter groups. We apply this to the representation theory of finite general linear groups G in nondescribing characteristic. In particular if we take as Mackey system the set of Levi subgroups of G,(considered as factor groups of the corresponding parabolic subgroups), we explain how vertices with respect to this Mackey system correspond to vertices in the associated Hecke algebra of type A. Using a version of Steinberg’s tensor product theorem in the nondescribing characteristic case the vertices of the irreducible representation with respect to the Mackey system of Levi subgroups (Harish-Chandra vertices) are described.

This is a clear, accessible and up-to-date exposition of modular representation theory of finite groups from a character-theoretic viewpoint. After a short review of the necessary background material, the early chapters introduce Brauer characters and blocks and develop their basic properties. The next three chapters study and prove Brauer's first, second and third main theorems in turn. These results are then applied to prove a major application of finite groups, the Glauberman Z*-theorem. Later chapters examine Brauer characters in more detail. The relationship between blocks and normal subgroups is also explored and the modular characters and blocks in p-solvable groups are discussed. Finally, the character theory of groups with a Sylow p-subgroup of order p is studied. Each chapter concludes with a set of problems. The book is aimed at graduate students, with some previous knowledge of ordinary character theory, and researchers studying the representation theory of finite groups.

Researchers in the fields of finite groups, representation theory, and algebra will appreciate this volume for it presents an excellent view of the many significant developments in the representation theory of finite groups. The papers in these proceedings were presented at the Summer Research Institute on Representations of Finite Groups, held at Humboldt State University in Arcata, California in July 1986. There has been intense research in the representation theory of finite groups in recent years.Particularly striking is the influence of algebraic geometry and cohomology theory in the modular representation theory and the character theory of reductive groups over finite fields, and in the general modular representation theory of finite groups. Noteworthy too are the continuing developments in block theory and the general character theory of finite groups. These proceedings contain nearly 100 papers of an expository and research nature and will give readers a sense of the new and important influences in this vital field.

Invertible map equivalences are approximations of graph isomorphism that refine the well-known Weisfeiler-Leman method. They are parameterized by a number k and a set Q of primes. The intuition is that two equivalent graphs G equiv^IM_{k, Q} H cannot be distinguished by means of partitioning the set of k-tuples in both graphs with respect to any linear-algebraic operator acting on vector spaces over fields of characteristic p, for any p in Q. These equivalences have first appeared in the study of rank logic, but in fact they can be used to delimit the expressive power of any extension of fixed-point logic with linear-algebraic operators. We define {LA^{k}}(Q), an infinitary logic with k variables and all linear-algebraic operators over finite vector spaces of characteristic p in Q and show that equiv^IM_{k, Q} is the natural notion of elementary equivalence for this logic. The logic LA^{omega}(Q) = Cup_{k in omega} LA^{k}(Q) is then a natural upper bound on the expressive power of any extension of fixed-point logics by means of Q-linear-algebraic operators. By means of a new and much deeper algebraic analysis of a generalized variant, for any prime p, of the CFI-structures due to Cai, Furer, and Immerman, we prove that, as long as Q is not the set of all primes, there is no k such that equiv^IM_{k, Q} is the same as isomorphism. It follows that there are polynomial-time properties of graphs which are not definable in LA^{omega}(Q), which implies that no extension of fixed-point logic with linear-algebraic operators can capture PTIME, unless it includes such operators for all prime characteristics. Our analysis requires substantial algebraic machinery, including a homogeneity property of CFI-structures and Maschke's Theorem, an important result from the representation theory of finite groups.

Orders in Hopf algebras

Chapter 1 - History of Measure Theory

In this chapter we provide an application of the structure theory of rings developed in Chapters One and Two to the theory of finite groups. Representation theory of finite groups is a vast subject; in this chapter we’ll make a thin beeline right to a famous theorem of Burnside. For a more thorough introduction to the representation theory of finite groups, the reader may consult Serre, Linear Representations of Finite Groups, as well as Fulton and Harris, Representation Theory : A First Course.

When we have a finite group we can identify the elements of the group as matrices over some field, so we can study the properties of the groups by studying the matrices only.Representation theory is very useful in number theory and to solve many group theorytical problems. In this thesis I have tried to show some useful applications representation theory of finite groups.

The representation theory of finite groups has seen rapid growth in recent years with the development of efficient algorithms and computer algebra systems. This is the first book to provide an introduction to the ordinary and modular representation theory of finite groups with special emphasis on the computational aspects of the subject. Evolving from courses taught at Aachen University, this well-paced text is ideal for graduate-level study. The authors provide over 200 exercises, both theoretical and computational, and include worked examples using the computer algebra system GAP. These make the abstract theory tangible and engage students in real hands-on work. GAP is freely available from www.gap-system.org and readers can download source code and solutions to selected exercises from the book's web page.

This appendix surveys those aspects of the representation theory of finite groups that are used throughout the text. The final section provides a brief overview of the representation theory of the symmetric group in characteristic zero. Monoid representation theory very much builds on group representation theory and so one should have a solid foundation in the latter subject before attempting to master the former. Good references for the representation theory of finite groups are [Isa76, Ser77, CR88]. See also [Ste12a]. We mostly omit proofs here, although we occasionally do provide a sketch or complete proof for convenience of the reader.

Local representation theory of transporter categories

D. J. Benson, Representations and cohomology, Vol. 1: Basic representation theory of finite groups and associative algebras (Cambridge Studies in Advanced Mathematics 30, Cambridge University Press1991), pp. xii + 224, 0 521 36134 6, £25. - Volume 34 Issue 3

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.