Theory of monomial groups

The present paper is the second part of [8] (brackets refer to bibliography) referred to below as Part I. We number sections, formulae, theorems, etc. consecutively from those in Part I, and use the same notation. For instance, f is a field of characteristic p with q (?_ cc ) elements; Zi is a t-vector space of dimension n; (5 = 65(n, f) is the full linear (modular) group of all f-automorphisms of 318 ; 93m is the space of all tensors of rank m over QI ; HIm is the Kronecker mth power representation of 6; 21m is the enveloping algebra of H1m. Superscripts zero indicate the analogous quantities defined over a field to of characteristic zero. The decomposition of non-modular tensors, or equivalently the determination of the reduced form of Ho is now a classical part of algebra. The objective of the present sequence of papers is to obtain a similar theory for modular tensors, or equivalently to study the reduced form of Hm . Any representation of 65 whose space is a subspace or factor space of 93m , or a direct sum of such spaces we call a tensor representation of 6. If q (and therefore 6) is finite we denote the group ring by r. One of the main results of the present paper is that there exists a faithful tensor representation of F (Th. X ?14). From this it follows from an unpublished theorem of Nesbitt that every representation of 6) is equivalent to a tensor representation, but we do not establish or apply this last result below. An important feature of the study of modular representations of groups has been the use of induced representations. One starts with a finite group and a non-modular representation of it, and after suitable number theoretic preparations take residue classes and obtain a modular representation. A generalization of this process would be to induce both the group and the representation. We leave for future investigation the determination of the general theory of such a process, and content ourselves here with the application of the idea to obtain from each irreducible representation of the non-modular full linear group a representation of the modular full linear group. This is done in ?9 below. In ?10 a character theory is developed for tensor representations of 6; this is applied in ?11 to prove that every irreducible representation of (M is equivalent to a tensor representation,1 and in ??12, 13 to obtain specific values of the irreducible and indecomposable modular characters for the representations H.m having m m. In ??15-20 the situation for m < 2p is cleaned up by re-

A subgroup H is called c -normal in a group G if there exists a normal subgroup N of G such that HN = G and H ∩ N ≤ H G , where H G ≕ Core( H ) is the maximal normal subgroup of G which is contained in H . We obtain the c -normal subgroups in symmetric and dihedral groups. Also we find the number of c -normal subgroups of order 2 in symmetric groups. We conclude by giving a program in GAP for finding c -normal subgroups. AMS Classification : 20D25. Keywords : c -normal, symmetric, dihedral. Introduction The relationship between the properties of maximal subgroups of a finite group G and the structure of G has been studied extensively. The normality of subgroups in a finite group plays an important role in the study of finite groups. It is well known that a finite group G is nilpotent if and only if every maximal subgroup of G is normal in G . In Wang introduced the concept of c -normality of a finite group. He used the c -normality of a maximal subgroup to give some conditions for the solvability and supersolvability of a finite group. For example, he showed that G is solvable if and only if M is c -normal in G for every maximal subgroup M of G . In this paper, we obtain the c -normal subgroups in symmetric and dihedral groups, and also we find the number of c -normal subgroups of order 2 in symmetric groups.

In chemistry, point group is a type of group used to describe the symmetry of molecules. It is a collection of symmetry elements controlled by a form or shape which all go through one point in space, which consists of all symmetry operations that are possible for every molecule. Next, a set of number or matrices which assigns to the elements of a group and represents the multiplication of the elements is said to constitute representation of a group. Here, each individual matrix is called a representative that corresponds to the symmetry operations of point groups, and the complete set of matrices is called a matrix representation of the group. This research was aimed to relate the symmetry in point groups with group theory in mathematics using the concept of isomorphism, where elements of point groups and groups were mapped such that the isomorphism properties were fulfilled. Then, matrix representations of point groups were found based on the multiplication table where symmetry operations were represented by matrices. From this research, point groups of order less than eight were shown to be isomorphic with groups in group theory. In addition, the matrix representation corresponding to the symmetry operations of these point groups wasis presented. This research would hence connect the field of mathematics and chemistry, where the relation between groups in group theory and point groups in chemistry were shown.

Matrices Algebras Groups The Frobenius algebra The symmetric group Immanants and $S$-functions $S$-functions of special series The calculation of the characters of the symmetric group Group characters and the structure of groups Continuous matrix groups and invariant matrices Groups of unitary matrices Appendix Bibliography Supplementary bibliography Index.

Chapter IV. The Symmetric Group and The Full Linear Group

This chapter opens Part Two, devoted to the study of transformation theory, with some additional topics in group theory that arise in musical applications. Transformation groups on finite spaces may be regarded as permutation groups; permutation groups on pitch-class space include not only the groups of transpositions and inversions but also the multiplication group, the affine group, and the symmetric group. Another musical illustration of permutations involves the rearrangement of lines in invertible counterpoint. The structure of a finite group may be represented in the form of a group table or a Cayley diagram (a kind of graph). Other concepts discussed include homomorphisms and isomorphisms of groups, direct-product groups, normal subgroups, and quotient groups. Groups underlie many examples of symmetry in music, as formalized through the study of equivalence relations, orbits, and stabilizers.

The Symmetric permutation group S2n is a classical algebraic object that is also used in Computer science, Coding theory, Statistics, etc. In particular, the coding theory considers codes defined on the symmetric group Sn or its subgroups. The research of permutation codes has been started from 1970s. These codes can be obtained with using different distances: Hamming, Ulam, Cailey, Levenshtein. The finding distance on permutations depends on their number of fixed or unfixed points. Therefore, it is natural to count the number of unfixed points in a certain group of permutations.In this paper, we consider the number of unfixed points of permutations that are elements of the Sylow 2-subgroup Syl2(S2n) of symmetric groups S2n. Leo Kaluzhnin used tables to represent the elements of these groups [8]. Volodymyr Nekrashevych represented permutations by their portraits [9]. We use algorithms that describe the connection between the permutation group Syl2(S2n) and the group of labeled binary rooted trees [10].An algorithm for finding the number of unfixed points for permutations of the Sylow 2-subgroup Syl2(S2n) of the symmetric group S2n is proposed in the article. An isomorphism between the group Syl2(S2n) and a group of labeled binary root trees was used to construct this algorithm. It is proved, that the algorithm of searching the number of unfixed point for permutations of the Sylow 2-subgroup Syl2(S2n) of the symmetric group S2n has complexity O(2n). In addition, the average number of steps of the algorithm for the Sylow 2-subgroup of the symmetric group S2n is found. The result for small n (n = 2, 3, 4) was verified with a program, that is written in the language of the computer algebra Sage. At the end of the article we find the number of permutations from Syl2(S2n ) that have a maximumnumber of unfixed points. The number of such permutations in the symmetric group S2n is well known.Obviously that this number is smaller for the Sylow 2-subgroup of the symmetric group Syl2(S2n ). Inthis case, we calculate the maximum number of unfixed points using a recursive formula.

(a 1, bl, ..., ag, bg; (al, bl)'..(ag, bg)= 1), where (al, bl)=ai bi ai-lbi -1. The mapping class group of the surface, Mg, has an algebraic characterization as a subgroup of the outer automorphism group of Hg. (Out (Hg) = Aut (Hg)/Inn (Hg).) Only those automorphism classes which are induced by free automorphisms on at,..., bg mapping (al, bl)..-(ag, bg) to a conjugate of itself rather than its inverse are permitted. In [2, p. 67] the question of whether Mg/M'g is trivial or of order 2 for g>2 is raised and the study of certain representations of Mg is suggested. In this paper we study some of these representations which we now describe. Let G and Q be arbitrary groups. We call N a Q-defining subgroup of G if N< G and G/N_~ Q. Aut (G) acts as a group of permutations on all Q-defining subgroups of G. Since Inn(G) acts trivially, Out(G) acts as a group of permutations on Q-defining subgroups of G/If, further, G is finitely generated and Q is finite we obtain a homomorphism from Out(G) to some 27 k (the symmetric group on k symbols). Letting G=Hg, we obtain permutation representations of MgcOUt(Hg). It is easily seen that if we choose for Q an abelian group, the representation obtained factors through the well known representation p: Mg--,PSp2g(7Z), the projective symplectic group, obtained by considering the action of Aut(Hg) on Hg/H'g and then factoring by the center. This is proved in Lemma 2 for the case Q=2~,, where it is shown that the action of Mg on 7Zn-defining subgroups is PSp2g(7Zn) as a permutation group on points of projective (2g-1)-space over 2g,. The main result of this paper is a description of the representation when Q =Dp, the dihedral group of order 2p, p an odd prime. The result holds true for De, where q is any square free odd integer, but for simplicity we limit our discussion to the case of odd primes. We will prove the following theorem. Theorem. The action of Mg on Dp-defining subgroups of Hg, for p an odd prime, is isomorphic to the full wreath product

We show that a large family of groups is uniformly stable relative to unitary groups equipped with submultiplicative norms, such as the operator, Frobenius, and Schatten p-norms. These include lamplighters Γ≀Λ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Gamma \\wr \\Lambda $$\\end{document} where Λ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Lambda $$\\end{document} is infinite and amenable, as well as several groups of dynamical origin such as the classical Thompson groups F,F′,T\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$F, F', T$$\\end{document} and V. We prove this by means of vanishing results in asymptotic cohomology, a theory introduced by the second author, Glebsky, Lubotzky and Monod, which is suitable for studying uniform stability. Along the way, we prove some foundational results in asymptotic cohomology, and use them to prove some hereditary features of Ulam stability. We further discuss metric approximation properties of such groups, taking values in unitary or symmetric groups.

Let G be a finite group with k conjugacy classes, and S(∞) be the infinite symmetric group, i.e., the group of finite permutations of {1, 2, 3, …}. Then the wreath product G ∞ = G ∼ S (∞) of G with S (∞) (called the big wreath product) can be defined. The group G ∞ is a generalization of the infinite symmetric group, and it is an example of a “big” group, in Vershik’s terminology. For such groups the two-sided regular representations are irreducible, the conventional scheme of harmonic analysis is not applicable, and the problem of harmonic analysis is a nontrivial problem with connections to different areas of mathematics and mathematical physics. Harmonic analysis on the infinite symmetric group was developed in the works by Kerov, Olshanski, and Vershik, and Borodin and Olshanski. The goal of this paper is to extend this theory to the case of G ∞. In particular, we construct an analogue $${\frak{S}}_{G}$$ S G of the space of virtual permutations. We then formulate and prove a theorem characterizing all central probability measures on $${\frak{S}}_{G}$$ S G . Next, we introduce generalized regular representations $$\{T_{z_{1}},\cdots,z_{k} : z_{1}\in \mathbb{C},\cdots,z_{k}\in \mathbb{C}\}$$ { T z 1 , ⋯ , z k : z 1 ∈ C , ⋯ , z k ∈ C } of the big wreath product G ∞, which are analogues of the Kerov–Olshanski–Vershik generalized regular representations of the infinite symmetric group. We derive an explicit formula for the characters of $$T_{z_{1}},\cdots,z_{k}$$ T z 1 , ⋯ , z k . The spectral measures of these representations are characterized in different ways. In particular, these spectral measures are associated with point processes whose correlation functions are explicitly computed. Thus, in representation-theoretic terms, the paper solves a natural problem of harmonic analysis for the big wreath products: our results describe the decomposition of $$T_{z_{1}},\cdots,z_{k}$$ T z 1 , ⋯ , z k into irreducible components.

Let X be the set of all natural numbers and let be the group of all finite permutations of X. The group equipped with the discrete topology, is called the infinite symmetric group. It was discussed in F. J. Murray and J. von Neumann as a concrete example of an ICC-group, which is a discrete group with infinite conjugacy classes. It is proved that the regular representation of an ICC-group is a factor representation of type II1. The infinite symmetric group is, therefore, a group not of type I. This may be the reason why its unitary representations have not been investigated satisfactorily. In fact, only few results are known. For instance, all indecomposable central positive definite functions on , which are related to factor representations of type IIl, were given by E. Thoma. Later on, A. M. Vershik and S. V. Kerov obtained the same result by a different method in and gave a realization of the representations of type II1 in. Concerning irreducible representations, A. Lieberman and G. I. Ol’shanskii obtained a characterization of a certain family of countably many irreducible representations by introducing a particular topology in However, irreducible representations have been studied not so actively as factor representations.

In this paper we consider a tree Lie algebra over a field of characteristic zero. This algebra is a module over the full linear group, and the spaces of homogeneous elements are invariant under this action. We study the decomposition of the homogeneous spaces into irreducible components and calculate their multiplicities. One method for calculating these multiplicities involves their connection with the values of the irreducible characters of the symmetric group on conjugacy classes of elements corresponding to a product of independent cycles of the same length. In the second section we give an explicit formula for calculating such character values. This formula is analogous to the hook formula for the dimension of the irreducible modules of the symmetric group. In the second method for calculating multiplicities we make use of Witt's formula for the dimensions of the polyhomogeneous components of a free Lie algebra. The rest of this paper deal with relations between the Hilbert series of a free two-generator Lie algebra and the generating series of the multiplicities of the irreducible modules in this algebra.

Part I. Quantum Groups, Representations of Groups and Algebras: On some problems in the theory of quantum groups by Ya. S. Soibelman and L. L. Vaksman Representations of quantum $*$-algebras $s1_t(N+1,\Bbb R)$ by E. Ye. Vaysleb Generalized Hall-Littlewood symmetric functions and orthogonal polynomials by S. V. Kerov Formulation of Bell type problems and noncommutative convex geometry by A. M. Vershik and B. S. Tsirelson Projective representations of the infinite symmetric group by M. L. Nazarov Structure theorems for a pair of unbounded selfadjoint operators satisfying a quadratic relation by V. L. Ostrovskii and Yu. S. Samoilenko Relations between compact quantum groups and Kac algebras by L. I. Vainerman Combinatorial complexity of orbits in representations of the symmetric group by A. I. Barvinok Part II. Dynamical Systems and Approximations: Adic models of ergodic transformations, spectral theory, substitutions, and related topics by A. M. Vershik and A. N. Livshits Periodic metrics by D. Yu. Burago Flows with positive entropy by D. Yu. Burago On the spectral theory of adic transformations by B. Solomyak On simultaneous action of Markov shift and adic transformation by M. Solomyak About a certain weakly mixing substitution by A. N. Livshits Manifolds with intrinsic metric, and nonholonomic spaces by V. N. Berestovskii and A. M. Vershik.

CHAPTER I - Foundations

Liftings of Projective 2-Dimensional Galois Representations and Embedding Problems