The ergodic theorem for random walks on finite quantum groups

By a finite quantum group, we will mean in this paper a finitedimensional Hopf algebra. A left Haar measure on such a quantum group is a linear functional satisfying a certain invariance property. In the theory of Hopf algebras, this is usually called an integral. It is well-known that, for a finite quantum group, there always exists a unique left Haar measure. This result can be found in standard works on Hopf algebras. In this paper we give a direct proof of the existence and uniqueness of the left Haar measure on a finite quantum group. We introduce the notion of a faithful functional and we show that the Haar measure is faithful. We consider the special case where the underlying algebra is a *-algebra with a faithful positive linear functional. Then the left and right Haar measures coincide. Finally, we treat an example of a root of unity algebra. It is an example of a finite quantum group where the left and right Haar measures are different. This note does not contain many new results but the treatment of the finite-dimensional case is very concise and instructive.

(2002). When Is a Group the Union of Proper Normal Subgroups? The American Mathematical Monthly: Vol. 109, No. 5, pp. 471-473.

A new characterisation of idempotent states on finite and compact quantum groups

Let H be a finite group having a fixed point free au tomorph ism c~ of order p". Consider the semidirect product G = (c~)H. It is well known that (eh) v" = 1 if h 9 H (see [3], p. 334). Put K = ( ev ) H. Then G # K and the elements in G K are p-elements. This last si tuation was considered by Kurzweil in [7]. It includes as a special case the groups having a proper generalized Hughes subgroup, i.e. those verifying G + Hr, (G) where Hp, (G) = ( x 9 G I xl" Je 1). A classical result of Hughes-Thompson and Kegel assures that if G :# H v (G) then H v (G) is ni lpotent (see [5] and [6]). Assuming that G is solvable Kurzweil showed that the Fi t t ing length of Hr, (G) (and hence that of G) is bounded by a function of n (see [7]). His bound for exceptional primes (in the Hal l -Higman sense) was improved by Har t ley and Rae as a product of their work in [4]. More recently Meixner obtained a l inear bound in [8]. Finally, in [2], the best possible bound f (Hr, (G)) < n was obtained for p odd. The case p = 2 is open. The purpose of this note is to consider the general problem. We may assume that G = ( x ) K, G K consists of p-elements and the order of x is, say, p". Assuming that G is solvable, what can be said about its Fi t t ing length? In [7] Kurzweil considered the case n = I and showed that f (K) < 2. Here we prove that f (K) < n + 1 if p is odd and the bound is best possible. The result is false for p = 2 even in the case n = 2. Our theorem is a new appl icat ion of the non-coprime Shult type theorems stated in [2]. There is another problem connected to this. Let G be a finite group having a proper subgroup H and a proper normal subgroup N of H such that H c~ H ~ < N if g 9 G H. Then G is said to be a Frobenius-Wie landt group (see [1] for more details and notation). We write (G, H, N) to indicate this situation. A theorem of Wielandt (see [1] for example) assures that, in such conditions, there exists a normal subgroup K of G such that G K = ~) (H -N) o, G = H K and H c~ K = N. Assume that H is a p-group. Then osG G K consists of p-elements. Thus we are in the above situation. Conversely, if G is p-solvable and K is a normal subgroup of G such that G K consists of p-elements then taking P 9 $1, (G) we have that (G, P, P c~ K) is an F W group. To show this observe that if x 9 G K then x acts f.p.f, on every x-invariant p '-section of K. Suppose that y 9 P c~ Po where g is a nontrivial p ' -element of G. As K is p-solvable we have a p '-section A/B of K where A and B are normal in G and g 9 A B. Then [y, g 1] 9 p c~ A < B. Thus y 9 P c~ K.

Finite quantum groups and quantum permutation groups

In this paper, we study convergence of random walks, on finite quantum groups, arising from linear combination of irreducible characters. We bound the distance to the Haar state and determine the asymptotic behavior, i.e., the limit state if it exists. We note that the possible limits are any central idempotent state. We also look at cutoff phenomenon in the Sekine finite quantum groups.

We establish a quantum version of Frucht's Theorem, proving that every finite quantum group is the quantum automorphism group of an undirected finite quantum graph. The construction is based on first considering several quantum Cayley graphs of the quantum group in question, and then providing a method to systematically combine them into a single quantum graph with the right symmetry properties. We also show that the dual of any non‐abelian finite group is ‘quantum rigid’. That is, always admits a quantum Cayley graph whose quantum automorphism group is exactly .

Coprime invariable generation and minimal-exponent groups

We present here the theory of quantum stochastic processes with independent increments with special emphasis on their structure as Markov processes. To avoid all technical difficulties we restrict ourselves to discrete time and finite quantum groups, i.e. finite-dimensional C*-Hopf algebras, see Appendix A. More details can be found in the lectures of Kummerer and Franz in this volume

A proper subgroup H of a group G is called a CC-subgroup of G if the centralizer C G (h) of h ∈ H # = H {1} is contained in H . Such finite groups were partially classified by G. F robenius , W. F eit , K. W. G ruenberg and O. H. K egel , J.S. W illiams , A. S.K ondrat'iev , N. I iyori and H.Y amaki , M. S uzuki , M. H erzog , Z. A rad , D. C hillag , C h . P raeger and others. In this report, using the classification of finite simple groups, we give a complete list of all finite groups containing a CC-subgroup. As a corollary we classify infinite profinite groups, locally finite groups and certain classes of topological groups containing a CC-subgroup under certain conditions. Introduction Let G denote a finite group. According to M. H erzog [18] a subgroup M ≤ G is a CC-subgroup (”centralizers contained“), if C G ( m ) ≤ M for every m ∈ M {1}. The example with smallest cardinality is G := S 3 with either M := 〈(123)〉 or M := 〈(12)〉 being a CC-subgroup. More generally, by the well known result of G. F robenius , every Frobenius group has CC-subgroups either the kernel or any complement. Sketching the thread One finds the concept of a CC-subgroup (without calling it that) in work of W. F eit describing doubly transitive groups which fix 3 letters (e.g. in [13]).

The general problem, with a particular instance of which the present paper is concerned, is to obtain a description of the local structure of a group from information about the global structure. The aspect of local structure investigated here is the embedding of subgroups, especially of nilpotent subgroups in finite soluble groups. A classification of embeddings of subgroups in finite groups by means of an arithmetic function called abnormal depth was proposed in [6]. Let H be a subgroup of a finite group G. Then a(G:H), the abnormal depth of H in G, is the least number of abnormal links appearing in any balanced chain of subgroups connecting H to G, that is a chain for which each link is either normal or abnormal. Thus a (G:H)= 0 if and only if H is subnormal in G; and a(G:P)__< 1 for every subgroup P of G of prime power order. It was shown in [6] that if H is a nilpotent subgroup of a finite soluble group G, of nilpotent length n, then a (G: H) =< n - 1. Here in w 1 we examine in greater detail the easiest non-trivial case, in which n = 2, and then in w 2 prove certain supplementary results for n = 3 and n = 4. Some simple wreath product properties are established in w 3 and used in w 4 for the construction of examples showing that the embedding results obtained cannot be improved in various obvious ways. Notation and terminology follow common usage. If t; and ~ are classes of groups, then 3s ~ denotes the class of all groups G having a normal subgroup X such that X e 3~ and G/X e ~. This defines a composition of classes of groups which in general is not associative. However, we shall deal only with classes of which the composition is associatNe, and we may therefore omit brackets from products of more than two classes. Since we shall be concerned exclusively with finite groups, we take 91 to denote the class of finite nilpotent groups and 9.1 the class of finite abelian groups. Then for any positive integer n, 9l" is the class of finite soluble groups of nilpotent lengths <__ n; and 9.I" is the class of finite soluble groups of derived lengths __< n. Henceforth the term group is understood to mean finite group. Then any group G has a unique smallest normal subgroup L such that G/L is nilpotent: G/L is called the 91-residual ofG. IfH is any subgroup of G, then there is a unique smallest normal subgroup of G containing H, called the normal closure of H in G and denoted by Ha; and a unique smallest subnormal subgroup of G containing H, called the subnormal closure of H in G and (following Wielandt [8]) denoted by H'" a. If H a = G, we shall say that H is contranormal in G. Then, for any subgroup H of G, it is clear that H is contranormal in H'" a. (This is to be compared with the fact that the hypernormalizer NE(H ) of H in G is self-normalizing in G.) An abnormal subgroup is both self-normalizing and

Let [Formula: see text] be a finite group. A proper subgroup [Formula: see text] of [Formula: see text] is said to be weakly monomial if the order of [Formula: see text] satisfies [Formula: see text]. In this paper, we determine all the weakly monomial maximal subgroups of finite simple groups.

Let G be a finite group. The spectrum of G is the set ω(G) of orders of all its elements. The subset of prime elements of ω(G) is denoted by π(G). The spectrum ω(G) of a group G defines its prime graph (or Grunberg-Kegel graph) Γ(G) with vertex set π(G), in which any two different vertices r and s are adjacent if and only if the number rs belongs to the set ω(G). We describe all the cases when the prime graphs of a finite simple group and of its proper subgroup coincide.

There are considered direct generalizations of finite Shmidt's groups, i.e., finite nonnilpotent groups, whose all proper subgroups are nilpotent. As corollaries, there are proved assertions confirming the dependence of the structure of the entire group on the presence of some system of Shmidt's groups. In particular, it is proved that a finite group is dispersive, if all its Shmidt's subgroups are upper-solvable.

A condition for distinguishing sceneries on non-abelian groups