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For finitely generated subgroups $W_1, \ldots , W_t$ of $\mathbb{Q}^{\times}$, integers $k_1, \ldots , k_t$, a Galois extension $F$ of $\mathbb{Q}$ and a union of conjugacy classes $C \subset \text{Gal}(F/\mathbb{Q})$, we develop methods for determining if there exists infinitely many primes $p$ such that the index of the reduction of $W_i$ modulo $p$ divides $k_i$ and such that the Artin symbol of $p$ on $F$ is contained in $C$. The results are a multivariable generalization of H.W. Lenstra's work. As an application, we determine all integers $a_1, \ldots , a_n$ such that $\text{ord}_p(a_1) = \ldots = \text{ord}_p(a_n)$ for infinitely many primes $p$. We also discuss the set of those $p$ for which $\text{ord}_p(a_1) > \ldots > \text{ord}_p(a_n)$. The obtained results are conditional to a generalization of the Riemann hypothesis., 20 pages. Add section on Kummer-type extensions and improve exposition

Let K K be an S n S_n -field. For a nonidentity conjugacy class C C , define N K , C N_{K,C} to be the smallest prime p p such that Frob p ∈ C _p\in C . By using the observation that N K , C N_{K,C} is interpreted as the first prime sign change of the Dirichlet coefficients of automorphic L L -functions, we improve the known bound on N K , C N_{K,C} for n = 3 , 4 , 5 n=3,4,5 . (For n = 5 n=5 , we need to assume the strong Artin conjecture.)

In this paper, we give a new and quick proof for the Hartman-Grobman theorem for random dynamical systems. This approach does not involve any previously proved existence of the stable and unstable manifolds.

How can we interpret the infimum of Lipschitz constants in a conjugacy class of interval maps? For positive entropy maps, the exponential of the topological entropy gives a well-known lower bound. We show that for piecewise monotone interval maps as well as for $C^{\infty}$ interval maps, these two quantities are equal, but for countably piecewise monotone maps, the inequality can be strict. Moreover, in the topologically mixing and Markov case, we characterize the infimum of Lipschitz constants as the exponential of the Salama entropy of a certain reverse Markov chain associated with the map. Dynamically, this number represents the exponential growth rate of the number of iterated preimages of nearly any point., Comment: 16 pages

In this paper we show that any Bohr-zero non-periodic set B B of traceless integer-valued matrices, denoted by Λ \Lambda , intersects non-trivially the conjugacy class of any matrix from Λ \Lambda . As a corollary, we obtain that the family of characteristic polynomials of B B contains all characteristic polynomials of matrices from Λ \Lambda . The main ingredient used in this paper is an equidistribution result for an S L d ( Z ) SL_d(\mathbb {Z}) random walk on a finite-dimensional torus deduced from Bourgain-Furman-Lindenstrauss-Mozes work [J. Amer. Math. Soc. 24 (2011), 231–280].

We show that every binary shift on the hyperfinite $II_1$ factor $R$ is cocycle conjugate to at least countably many non-conjugate binary shifts. This holds in particular for binary shifts of infinite commutant index., Comment: 6 pages

In a recent paper, Duncan, Griffin and Ono provide exact formulas for the coefficients of McKay-Thompson series and use them to find asymptotic expressions for the distribution of irreducible representations in the moonshine module V ♮ = ⨁ n V n ♮ V^\natural = \bigoplus _n V_n^\natural . Their results show that as n n tends to infinity, V n ♮ V_n^\natural is dominated by direct sums of copies of the regular representation. That is, if we view V n ♮ V_n^\natural as a module over the group ring Z [ M ] \mathbb {Z}[\mathbb {M}] , the free part dominates. A natural problem, posed at the end of the aforementioned paper, is to characterize the distribution of irreducible representations in the non-free part. Here, we study asymptotic formulas for the coefficients of McKay-Thompson series to answer this question. We arrive at an ordering of the series by the magnitude of their coefficients, which corresponds to various contributions to the distribution. In particular, we show how the asymptotic distribution of the non-free part is dictated by the column for conjugacy class 2A in the monster’s character table. We find analogous results for the other monster modules V ( − m ) V^{(-m)} and W ♮ W^\natural studied by Duncan, Griffin, and Ono.

For Anosov flows obtained by suspensions of Anosov diffeomorphisms on surfaces, we show the following type of rigidity result: if a topological conjugacy between them is differentiable at a point, then the conjugacy has a smooth extension to the suspended 3-manifold. These result generalize the similar ones of Sullivan and Ferreira-Pinto for 1-dimensional expanding dynamics and also a result of Ferreira-Pinto for 2-dimensional hyperbolic dynamics., 8 pages, 2 figures

We give quadratic bounds on the dimension of the space of conjugacy classes of subgroups of S L n ( R ) SL_n(\mathbb {R}) that are limits under conjugacy of the diagonal subgroup. We give the first explicit examples of abelian n − 1 n-1 -dimensional subgroups of S L n ( R ) SL_n(\mathbb {R}) which are not such a limit, and show that all such abelian groups are limits of the diagonal group iff n ≤ 4 n \leq 4 .

We investigate two "categorified" braid conjugacy class invariants, one coming from Khovanov homology and the other from Heegaard Floer homology. We prove that each yields a solution to the word problem but not the conjugacy problem in the braid group.

For a topological dynamical system consisting of a continuous map f, and a (not necessarily compact) subset Z of X, Bowen (1973) defined a dimension-like version of entropy, h_X(f,Z). In the same work, he introduced a notion of entropy-conjugacy for pairs of invertible compact systems: the systems (X,f) and (Y,g) are entropy-conjugate if there exist invariant Borel subsets X' of X and Y' of Y such that h_X(f,X\setminus X') < h_X(f,X), h_Y(g,Y \setminus Y') < h_Y(g,Y), and (X',f|_{X'}) is topologically conjugate to (Y',g|_{Y'}). Bowen conjectured that two mixing shifts of finite type are entropy-conjugate if they have the same entropy. We prove that two mixing shifts of finite type with equal entropy and left ideal class are entropy-conjugate. Consequently, in every entropy class Bowen's conjecture is true up to finite index.

Let $K$ be an algebraically closed field of characteristic 0 and $f\in K[t]$ a polynomial of degree $d\geq 2$. There exists a local conjugacy $\psi_f(t)\in tK[[1/t]]$ such that $\psi_f(t^d)=f(\psi_f(t))$. It has been known that $\psi_f$ is transcendental over $K(t)$ if $f$ is not conjugate to $t^d$ or a constant multiple of the Chebyshev polynomial. In this paper, we study the algebraic independence of $\psi_{f_1}$,\ldots,$\psi_{f_n}$ using a recent result of Medvedev-Scanlon. Related questions in transcendental number theory and canonical heights in arithmetic dynamics are also discussed.

We extend some results of Marmi--Moussa--Yoccoz on the cohomological equations and local conjugacy classes of interval exchange maps of restricted Roth type. In particular, we answer a question of Krikorian about the codimension of the local conjugacy class of self-similar interval exchange maps associated to the Eierlegende Wollmilchsau and the Ornithorynque., 14 pages, no figures, final version based on the referee report. To appear in Proc. Amer. Math. Soc

Let k((t)) be the field of formal Laurent power series over some field k of characteristic zero, q ∈ k which is not a root of unity, and σq be the automorphism of k((t)) defined by σq : α(t) 7→ α(qt). For an algebraic group G over k write G(k((t))) for the k((t))-points of G. Then σq also acts on G(k((t))) by change of variable, namely σq(g(t)) = g(qt). One can define a q-conjugate action of G(k((t))) on itself by

We give upper bounds for the number of irreducible representations of dimension at most n for a compact semisimple Lie group. In particular, we prove that there are at most n irreducible representations of dimension at most n for a simple compact Lie group. We use this prove that the number of conjugacy classes of maximal subgroups of a compact simple Lie group is O(r) where r is the Lie rank. We also give a short proof that the dimensions of the weight spaces of a maximal torus are small relative to the dimension of an irreducible module.

Using probabilistic methods, Collins and Dykema proved that the free product of two sofic groups amalgamated over a monotileably amenable subgroup is sofic as well. We show that the restriction is unnecessary; the free product of two sofic groups amalgamated over an arbitrary amenable subgroup is sofic. We also prove a group theoretical analogue of a result of Kenley Jung. A finitely generated group is amenable if and only if it has only one sofic representation up to conjugacy equivalence., Comment: Reference added

It is shown that if G G is a group of even order with trivial center such that | G | > 2 | C G ( t ) | 3 |G|>2|C_{G}(t)|^{3} for some involution t ∈ G t\in G , then there exists a proper subgroup H H of G G such that | G | > | H | 2 |G|> |H|^{2} . If | G | > | C G ( t ) | 3 |G|>|C_{G}(t)|^{3} and k ( G ) k(G) is the class number of G G , then | G | ≤ k ( G ) 3 |G|\leq k(G)^{3} .

In the paper, new criterions for existence and conjugacy of Hall subgroups of finite groups are given. In particular, the Schur-Zassenhaus theorem, Hall theorem and Cunihin theorem are generalized.

We study Klyachko models of SL(n, F), where F is a non-Archimedean local field. In particular, using results of Klyachko models for GL(n, F) due to Heumos, Rallis, Offen and Sayag, we give statements of existence, uniqueness, and disjointness of Klyachko models for admissible representations of SL(n, F), where the uniqueness and disjointness are up to specified conjugacy of the inducing character, and the existence is for unitarizable representations in the case F has characteristic 0. We apply these results to relate the size of an L-packet containing a given representation of SL(n, F) to the type of its Klyachko model, and we describe when a self-dual unitarizable representation of SL(n, F) is orthogonal and when it is symplectic.

In this paper we study the equivalence relation on the set of acyclic orientations of a graph Y that arises through source-to-sink conversions. This source-to-sink conversion encodes, e.g. conjugation of Coxeter elements of a Coxeter group. We give a direct proof of a recursion for the number of equivalence classes of this relation for an arbitrary graph Y using edge deletion and edge contraction of non-bridge edges. We conclude by showing how this result may also be obtained through an evaluation of the Tutte polynomial as T(Y,1,0), and we provide bijections to two other classes of acyclic orientations that are known to be counted in the same way. A transversal of the set of equivalence classes is given., Added a few results about connections to the Tutte polynomial

Many authors, e.g. M. Atiyah, R. Bott, J. Milnor, G. Bredon, S. Cappell, J. Shaneson, C. Sanchez, T. Petrie, E. Laitinen, K. Pawalowski, R. Solomon and so on, studied Smith equivalent representations for finite groups. Observing their results, E. Laitinen conjectured that nonisomorphic Smith equivalent real G-modules exist if a G, the number of real conjugacy classes of elements not of prime power order in G, is greater than or equal to 2. This paper shows that in the case G = Aut(A 6 ), a G = 2 any two Smith equivalent real G-modules are isomorphic.

We show that if G G is a finite group, C C a conjugacy class of G G and d = | C | d=\left \vert C\right \vert , d 2 , d 3 , … , d m d_{2},d_{3},\ldots ,d_{m} are the distinct elements in the multiset { | C | χ ( C ) χ ( 1 ) | χ ∈ I r r ( G ) } \left \{ \frac {\left \vert C\right \vert \chi (C)} {\chi (1)}\ |\ \chi \in \mathrm {Irr}(G)\right \} (here χ ( C ) \chi (C) is the value of χ \chi on any element of C C ), then \[ | G / ⟨ C ⟩ | ⋅ ( d − d 2 ) ( d − d 3 ) ⋯ ( d − d m ) ≡ 0 mod ⁡ | G | . \left \vert G/\left \langle C\right \rangle \right \vert \cdot \left ( d-d_{2}\right ) \left ( d-d_{3}\right ) \cdots \left ( d-d_{m}\right ) \equiv 0\ \operatorname {mod}\ \left \vert G\right \vert . \] This is a dual to a generalization of a theorem of Blichfeldt stating that if G G is a finite group, θ \theta a generalized character and d = θ ( 1 ) , d 2 , d 3 , … , d m d=\theta (1),d_{2},d_{3},\ldots ,d_{m} are the distinct values of θ \theta , then \[ | ker ⁡ ( θ ) | ( d − d 2 ) ( d − d 3 ) ⋯ ( d − d m ) ≡ 0 mod ⁡ | G | . \left \vert \ker (\theta )\right \vert \left ( d-d_{2}\right ) \left ( d-d_{3}\right ) \cdots \left ( d-d_{m}\right ) \equiv 0\ \operatorname {mod} \ \left \vert G\right \vert . \] We also observe that d = θ ( 1 ) d=\theta (1) in Blichfeldt’s congruence can be replaced, with a minor adjustment, by any rational value of θ \theta . A similar change can be done to the first congruence above.

In this paper, we will consider (germs of) holomorphic mappings of the form (f(z), λ 1 w 1 (1 + g 1 (z)),...,λ n w n (1 + g n (z))), defined in a neighborhood of the origin in C n+1 . Most of our interest is in those mappings where f(z) = z + a m z m +... is a germ tangent to the identity and g i (0) = 0 for i = 1,..., n, and i i 6 C possess no resonances, for these are the so-called Poincare-Dulac normal forms of the mappings (z + 0(2), λ 1 ω + O(2) λ n ω + 0(2)). We construct formal normal forms for these mappings and discuss a condition which tests for the convergence or divergence of the conjugating maps, giving specific examples.

A self-map f f on the CW complex Z Z is a periodic homotopy idempotent if for some r ⩾ 0 r\geqslant 0 and p > 0 p>0 the iterates f r f^r and f r + p f^{r+p} are homotopic. Geoghegan and Nicas defined the rotation index R I ( f ) RI(f) of such a map. They proved that for r = p = 1 r=p=1 , the homotopy idempotent f f splits if and only if R I ( f ) = 1 RI(f)=1 , while for r = 0 r=0 , the index R I ( f ) RI(f) divides p 2 p^2 . We extend this to arbitrary p p and r r , and generalize various results related to the splitting of homotopy idempotents on CW complexes and conjugacy idempotents on groups.

We show that any pointed Hopf algebra with infinitesimal braiding associated to the conjugacy class of π ∈ S n \pi \in \mathbb {S}_n is infinite-dimensional, if either the order of π \pi is odd, or all cycles in the decomposition of π \pi as a product of disjoint cycles have odd order except for exactly two transpositions.

We show that if H is a reductive group, then nth roots of conjugacy classes are a finite union of conjugacy classes, and that if G is an algebraic overgroup of H, then the intersection of H with a conjugacy class of G is a finite union of H-conjugacy classes. These results follow from results on finiteness of unipotent classes in an almost simple algebraic group.

Suppose that G is a …nite solvable group which has an irreducible characterwhich vanishes on exactly one conjugacy class. Then we show that G has a homomorphic image which is a nontrivial 2-transitive permutation group. The latter groups have been classi…ed by Huppert. We can also say more about the structure of G depending on whether � is primitive or not. Mathematics Subject Classi…cation 2000: 20C15 20D10 20B20

Let the Ricci curvature of a compact Riemannian manifold be greater, at every point, than the Lie derivative of the metric with respect to some fixed smooth vector field. It is shown that the fundamental group then has only finitely many conjugacy classes. This applies, in particular, to all compact shrinking Ricci solitons., Comment: 4 pages

We prove that knowledge of the character degrees of a finite group G and of their multiplicities determines whether G has a Sylow p-subgroup as a direct factor. An analogous result based on knowledge of the conjugacy class sizes is known. We prove variations of both results and discuss their similarities

In this paper we prove that the general version F ( N ) F(N) of the Thompson group is inner amenable. As a consequence we generalize a result of P. Jolissaint. To do so, we prove first that F ( N ) F(N) together with a normal subgroup are i.c.c (infinite conjugacy classes) groups. Then, we investigate the relative McDuff property out of which we extract property Γ \Gamma for the group von Neumann algebras involved. By a result of E. G. Effros, F ( N ) F(N) follows inner amenable.

We prove two theorems linking the cohomology of a pro-p group G with the conjugacy classes of its finite subgroups. The number of conjugacy classes of elementary abelian p-subgroups of G is finite if and only if the ring H*(G, Z/p) is finitely generated modulo nilpotent elements. If the ring H*(G,Z/p) is finitely generated, then the number of conjugacy classes of finite subgroups of G is finite.

We prove that on R 3 , except for those germs of vector fields whose linear parts are conjugated to λx∂/∂x + λy∂/∂y + 2λz∂/∂z, any two Poincare type vector fields are at least C 3 conjugated to each other provided their linear approximations have the same eigenvalues and the nonlinear parts are generic.

For orders and conjugacy in finite group theory, Lagrange’s Theorem and the class equation have universal application. Here, the class equation (extended to monoids via standard group action by conjugation) is applied to factorizable submonoids of the symmetric inverse monoid. In particular, if M M is a monoid induced by a subgroup G G of the symmetric group S n S_n , then the center Z G ( M ) Z_G(M) (all elements of M M that commute with every element of G G ) is Z ( G ) ∪ { 0 } Z(G) \cup \{0\} if and only if G G is transitive. In the case where G G is both transitive and of order either p p or p 2 p^2 (for p p prime), formulas are provided for the order of M M as well as the number and sizes of its conjugacy classes.

The automorphism group of a free group Ant(F k ) acts on the set of generating k-tuples (g 1 ,...,g k ) of a group G. Higman showed that when k = 2, the union of conjugacy classes of the commutators [g 1 , g 2 ] and [g 2 , g 1 ] is an orbit invariant. We give a negative answer to a question of B.H. Neumann, as to whether there is a generalization of Higman's result for k > 3.

In this paper, we bound the Fitting height of a solvable group by the number of zeros in a character table.

In this note we prove that for any two integers r, s > 1 there exist finite p-groups G of class 2 such that | cd(G)| = r and | cs(G)| = s.

In 1992, Blanc and Brylinski showed the following property for a p p -adic group G G , called the “abstract Selberg principle”: the orbital integrals on conjugacy classes of non-compact elements of the Hattori rank of a finitely generated projective smooth representation of G G vanish. The proof is by explicit computations of “low” level ( 0 0 and 1 ) 1) cyclic and Hochschild cohomologies. Here we intend to show that this property is actually a direct consequence of two facts: Clozel’s integration formula (which leads us to assume the defining characteristic to be zero) and the triviality of the action of unramified characters on the K 0 K_0 of G G (which is also proven here, using a standard K K -theoretic argument due to Grothendieck).

We show that the converse to the main theorem of Ergodic transformations conjugate to their inverses by involutions, by Goodson et al. (Ergodic Theory and Dynamical Systems 16 (1996), 97–124), holds in the unitary category. Specifically it is shown that if U U is a unitary operator defined on an L 2 L^{2} space which preserves real valued functions, and if U − 1 S = S U U^{-1}S=SU implies S 2 = I S^{2}=I whenever S S is another such operator, then U U has simple spectrum. The corresponding result for measure preserving transformations is shown to be false. The counter-example we have involves Gaussian automorphisms. We show that a Gaussian automorphism is always conjugate to its inverse, so that the Inverse-Conjugacy Theorem is applicable to such maps having simple spectrum. Furthermore, there are Gaussian automorphisms having non-simple spectrum for which every conjugation of T T with T − 1 T^{-1} is an involution.

This work concerns a generalization of Clifford theory to blocks of group-graded algebras. A module-theoretic approach is taken to prove a one-to-one correspondence between the blocks of a fully group-graded algebra covering a given block of its identity component, and conjugacy classes of blocks of a twisted group algebra. In particular, this applies to blocks of a finite group covering blocks of a normal subgroup.

Using the fundamental group of a punctured torus, a free group F of rank two, and the fact that the natural eipmorphism from AutF onto Aut(F/F') has as kernel the group of inner automorphisms of F, we describe representatives of the conjugacy classes of generating pairs of F and give explicit relations between them. Let F = F(S, T) be the free group on S and T. By a theorem of Nielsen [N] (see [LS, p. 25]) the natural epimorphism from AutF onto Aut(F/F') ( = GL(2, Z)) has as kernel the group of inner automorphisms of F. From this it follows easily that, if a is the abelianization homomorphism from F onto F/F' (= Z2) and a E 22 is primitivel, then the inverse image of a under a is a conjugacy class of primitive elements. Also, if (ai,a2) is a basis of 22, then, up to conjugacy, there is a unique basis (fi,f2) of F such that (fi)a = ai (i = 1,2). (The basis (fi, f2) is conjugate to (gl, 2) if there exists w E F such that w-lfiw = gi (i = 1,2)). In the important paper [OZ], Osborne and Zieschang define explicitly primitive words Wm,n E F(S,T), where m and n are relatively prime integers, such that (Wm,n)a = (m, n). They also state that if mn pq = 1, then (Wm,n, Wp,q) is a basis of F; this, while correct for nonnegative values of m,n,p, q, is not valid in general (for example W-2,-3 and W1,1 do not generate F). A composition formula is also stated in [OZ, Thm. 3.5] but this, even with the correction of indices in [LTZ, 2.1.3], is incorrect in general. In the present article we consider elements Va of F for a = (m, n) E Z2 and e E D C IR2 where D is the complement of the union of all the lines that intersect 22 in more than one point. If gcd(m,n) = 1, then V(,) is conjugate to Wm,n. We show in Theorem 1.i) that (Va, Vb) is a basis of F, if Z2 = (a, b), and obtain in Theorem l.ii) a composition formula. Everything is obtained by applying the fundamental group functor ir to the punctured torus. Denote by T the torus IR2/ 22, by To the punctured torus (R2 _ Z2) / Z2 and by p : R2 Z2 -To the natural projection. If a E 22 and e E D, then denote (e)p by E and define ya E 7r(To, ) as the homotopy class of the loop (e+ta)p, t E [0,1]. Denote ' o0) (resp. y0 ,1)) by S (resp. Te). There is an isomorphism Received by the editors September 15, 1997. 1991 Mathematics Subject Classification. Primary 57M07, 20E05.

A one-dimensional shift of finite type can be described as the collection of bi-infinite “walks" along an edge graph. The Decomposition Theorem states that every conjugacy between two shifts of finite type can be broken down into a finite sequence of splittings and amalgamations of their edge graphs. When dealing with two-dimensional shifts of finite type, the appropriate edge graph description is not as clear; we turn to Nasu’s notion of a “textile system" for such a description and show that all two-dimensional shifts of finite type can be so described. We then define textile splittings and amalgamations and prove that every conjugacy between two-dimensional shifts of finite type can be broken down into a finite sequence of textile splittings, textile amalgamations, and a third operation called an inversion.

We present several results connecting the number of conjugacy classes of a finite group on which an irreducible character vanishes, and the size of some centralizer of an element. For example, we show that if G G is a finite group such that G ≠ G ′ ≠ G G\ne G’\ne G , then G G has an element x x , such that | C G ( x ) | ≤ 2 m |C_G(x)|\le 2m , where m m is the maximal number of zeros in a row of the character table of G G . Dual results connecting the number of irreducible characters which are zero on a fixed conjugacy class, and the degree of some irreducible character, are included too. For example, the dual of the above result is the following: Let G G be a finite group such that 1 ≠ Z ( G ) ≠ Z 2 ( G ) 1\ne Z(G)\ne Z_2(G) ; then G G has an irreducible character χ \chi such that | G | χ 2 ( 1 ) ≤ 2 m \frac {|G|}{\chi ^2(1)}\le 2m , where m m is the maximal number of zeros in a column of the character table of G G .

Using smooth one-fixed-point actions on spheres and a result due to Bob Oliver on the tangent representations at fixed points for smooth group actions on disks, we obtain a similar result for perfect group actions on spheres. For a finite group G, we compute a certain subgroup IO'(G) of the representation ring RO(G). This allows us to prove that a finite perfect group G has a smooth 2-proper action on a sphere with isolated fixed points at which the tangent representations of G are mutually nonisomorphic if and only if G contains two or more real conjugacy classes of elements not of prime power order. Moreover, by reducing group theoretical computations to number theory, for an integer n > 1 and primes p, q, we prove similar results for the group G = AnX SL2 (Fp), or PSL2 (Fq). In particular, G has Smith equivalent representations that are not isomorphic if and only if n > 8, p > 5, q > 19.

Let M(G) be the subgroup of G generated by all elements that lie in conjugacy classes of the two smallest sizes. Avinoam Mann showed that if G is nilpotent, then M(G) has nilpotence class at most 3. Using a slight variation on Mann's methods, we obtain results that do not require us to assume that G is nilpotent. We show that if G is supersolvable, then M(G) is nilpotent with class at most 3, and in general, the Fitting subgroup of M(G) has class at most 4.