Conjugacy Classes Research Articles

The average conjugacy class size and the Hall π-subgroups in π-solvable groups

Let [Formula: see text] be a [Formula: see text]-solvable group for some set of primes [Formula: see text]. We show that if [Formula: see text] is a Hall [Formula: see text]-subgroup of [Formula: see text], then [Formula: see text], where [Formula: see text] is the number of conjugacy classes of [Formula: see text].

Mapping classes are almost determined by their finite quotient actions

Given any connected compact orientable surface, a pair of mapping classes is said to be procongruently conjugate if it induces a conjugate pair of outer automophisms on the profinite completion of the fundamental group of the surface. For example, this occurs if they induce conjugate outer automorphisms on every characteristic finite quotient of the fundamental group. In this paper, it is shown that every procongruent conjugacy class of mapping classes, as a subset of the surface mapping class group, is the disjoint union of at most finitely many conjugacy classes of mapping classes. For any pseudo-Anosov mapping class of a connected closed orientable surface, several topological features are confirmed to depend only on the procongurent conjugacy class of the mapping class, including the stretching factor, the topological type of the prong singularities, the transverse orientability of the invariant foliations, and the isomorphism type of the symplectic Floer homology.

Groups whose vanishing class sizes are p-powers

–Given a finite group G, an element is called a vanishing element of G if it is a zero of an irreducible character of G; otherwise, it is called a non-vanishing element. Moreover, the conjugacy class of an element is termed as a vanishing class if that element is a vanishing element. In this paper, we describe the finite groups whose vanishing class sizes are all powers of a prime p.

Lengths of conjugacy classes of π-elements and primary elements in finite groups

In this paper, we study how certain arithmetical conditions on the lengths of conjugacy classes of primary elements and π-elements of a finite group G influence the structure of G, some new results are obtained and several well-known results in the literature are developed. In particular, the structure of a kind of outer p-supersolvable groups is described, some new conditions implying π-supersolvability are found, and the upper bound for the order of Sylow subgroups is discussed.

Conjugacy classes of small size in solvable groups

If G is a finite group and , we say that x lies in a small class if is minimal among the sizes of the noncentral conjugacy classes of G. It has been conjectured that if G is a solvable group with trivial center and x belongs to a small class, then x lies in the center of the Fitting subgroup of G. We restrict the structure of a possible counterexample to this conjecture. We discuss the possible existence of a counterexample. As a consequence, we prove the conjecture when the small classes have prime sizes and also when all chief factors of G have rank at most 2. Perhaps surprisingly, the proof in the case when the small classes have prime size and the discussion on the existence of counterexamples use techniques from linear algebra.

A solution to Brauer's Problem 14

It is well known that the number of real irreducible characters of a finite group G coincides with the number of real conjugacy classes of G. Richard Brauer has asked if the number of irreducible characters with Frobenius–Schur indicator 1 can also be expressed in group theoretical terms. We show that this can done by counting solutions of g12…gn2=1 with g1,…,gn∈G.

Conjugacy classes of maximal cyclic subgroups

Abstract In this paper, we study the number of conjugacy classes of maximal cyclic subgroups of a finite group 𝐺, denoted η ⁢ ( G ) \eta(G) . First we consider the properties of this invariant in relation to direct and semi-direct products, and we characterize the normal subgroups 𝑁 with η ⁢ ( G / N ) = η ⁢ ( G ) \eta(G/N)=\eta(G) . In addition, by applying the classification of finite groups whose nontrivial elements have prime order, we determine the structure of G / ⟨ G − ⟩ G/\langle G^{-}\rangle , where G − G^{-} is the set of elements of 𝐺 generating non-maximal cyclic subgroups of 𝐺. More precisely, we show that G / ⟨ G − ⟩ G/\langle G^{-}\rangle is either trivial, elementary abelian, a Frobenius group or isomorphic to A 5 A_{5} .

Lengths of involutions in finite Coxeter groups

Let W be a finite Coxeter group and X a subset of W . The length polynomial L W , X ( t ) is defined by L W , X ( t ) = ∑ x ∈ X t ℓ ( x ) , where ℓ is the length function on W . If X = { x ∈ W : x 2 = 1 } then we call L W , X ( t ) the involution length polynomial of W . In this article we derive expressions for the length polynomial where X is any conjugacy class of involutions, and the involution length polynomial, in any finite Coxeter group W . In particular, these results correct errors in [11] for the involution length polynomials of Coxeter groups of type B n and D n . Moreover, we give a counterexample to a unimodality conjecture stated in [11] .

Influence of Conjugacy Class Sizes of Some Elements on the Structure of a Finite Group

Influence of Conjugacy Class Sizes of Some Elements on the Structure of a Finite Group

Projective characters of metacyclic p-groups

Let ω be a 2-cocycle of a metacyclic p-group G representing a non-trivial element of the Schur multiplier Then the number of ω-regular conjugacy classes of G, the subgroup consisting of the ω-regular elements in the center of G, the degree of each irreducible ω-character of G and a representation group H of G with M(H) trivial are all determined. Finally, for ω constructed from H, the projective character table of G corresponding to ω is found in the case that G is of positive type. Communicated by Mark Lewis

The R ∞ R_{\infty} property for nilpotent quotients of Generalized Solvable Baumslag–Solitar groups

Abstract We say a group 𝐺 has property R ∞ R_{\infty} if the number R ⁢ ( φ ) R(\varphi) of twisted conjugacy classes is infinite for every automorphism 𝜑 of 𝐺. For such groups, the R ∞ R_{\infty} -nilpotency degree is the least integer 𝑐 such that G / γ c + 1 ⁢ ( G ) G/\gamma_{c+1}(G) has property R ∞ R_{\infty} . In this work, we compute the R ∞ R_{\infty} -nilpotency degree of all Generalized Solvable Baumslag–Solitar groups Γ n \Gamma_{n} . Moreover, we compute the lower central series of Γ n \Gamma_{n} , write the nilpotent quotients Γ n , c = Γ n / γ c + 1 ⁢ ( Γ n ) \Gamma_{n,c}=\Gamma_{n}/\gamma_{c+1}(\Gamma_{n}) as semidirect products of finitely generated abelian groups and classify which invertible integer matrices can be extended to automorphisms of Γ n , c \Gamma_{n,c} .

On genera containing non-split Eichler orders over function fields

Grothendieck-Birkhoff Theorem states that every finite dimensional vector bundle over the projective line P1 splits as the sum of one dimensional vector This can be rephrased, in terms of orders, as stating that all maximal orders over the projective line in a matrix algebra split. In this work we study the extent to which this result can be generalized to Eichler orders when the base field F is finite. To be precise, we characterize both the genera of Eichler orders containing only split orders and the genera containing only a finite number of non-split conjugacy classes. The latter characterization is given for arbitrary projective curves over F. The method developed here also allows us to compute quotient graphs for some subgroups of $PGL_2(F[t])$ of arithmetical interest. This paper includes material from the unpublished work Simultaneous diagonalization of vector bundles.

A parametrization of sheets of conjugacy classes in bad characteristic

AbstractLet G be a simple algebraic group of adjoint type over an algebraically closed field k of bad characteristic. We show that its sheets of conjugacy classes are parametrized by G-conjugacy classes of pairs $(M,{\mathcal O})$ where M is the identity component of the centralizer of a semisimple element in G and ${\mathcal O}$ is a rigid unipotent conjugacy class in M, in analogy with the good characteristic case.

On Whitehead’s cut vertex lemma

Abstract One version of Whitehead’s famous cut vertex lemma says that if an element of a free group is part of a free basis, then a certain graph associated to its conjugacy class that we call the star graph is either disconnected or has a cut vertex. We state and prove a version of this lemma for conjugacy classes of elements and convex-cocompact subgroups of groups acting cocompactly on trees with finitely generated edge stabilizers.

Relative hyperbolicity of hyperbolic-by-cyclic groups

Let G be a torsion-free hyperbolic group and \alpha an automorphism of G . We show that there exists a canonical collection of subgroups that are polynomially growing under \alpha , and that the mapping torus of G by \alpha is hyperbolic relative to the suspensions of the maximal polynomially growing subgroups under \alpha . As a consequence, we obtain a dichotomy for growth: given an automorphism of a torsion-free hyperbolic group, the conjugacy class of an element either grows polynomially under the automorphism, or at least exponentially.

Torsion in 1-Cusped Picard Modular Groups

We present a systematic effective method to construct coarse fundamental domains for the action of the Picard modular groups $PU(2,1,\mathcal {O}_{d})$ where $\mathcal {O}_{d}$ has class number one, i.e., d = 1,2,3,7,11,19,43,67,163. The computations can be performed quickly up to the value d = 19. As an application of this method, we classify conjugacy classes of torsion elements, deduce short presentations for the groups, and construct neat subgroups of small index.

On the sharp Baer-Suzuki theorem for the $\pi$-radical of a finite group

Let $\pi$ be a proper subset of the set of prime numbers. Denote by $r$ the least prime not contained in $\pi$ and set $m=r$ for $r=2$ and $3$ and $m=r-1$ for $r\geqslant5$. The conjecture under consideration claims that a conjugacy class $D$ of a finite group $G$ generates a $\pi$-subgroup of $G$ (equivalently, is contained in the $\pi$-radical) if and only if any $m$ elements of $D$ generate a $\pi$-group. It is shown that this conjecture holds if every non-Abelian composition factor of $G$ is isomorphic to a sporadic, an alternating, a linear, or a unitary simple group. Bibliography: 49 titles.

Double coset Markov chains

AbstractLetGbe a finite group. Let$H, K$be subgroups ofGand$H \backslash G / K$the double coset space. IfQis a probability onGwhich is constant on conjugacy classes ($Q(s^{-1} t s) = Q(t)$), then the random walk driven byQonGprojects to a Markov chain on$H \backslash G /K$. This allows analysis of the lumped chain using the representation theory ofG. Examples include coagulation-fragmentation processes and natural Markov chains on contingency tables. Our main example projects the random transvections walk on$GL_n(q)$onto a Markov chain on$S_n$via the Bruhat decomposition. The chain on$S_n$has a Mallows stationary distribution and interesting mixing time behavior. The projection illuminates the combinatorics of Gaussian elimination. Along the way, we give a representation of the sum of transvections in the Hecke algebra of double cosets, which describes the Markov chain as a mixture of Metropolis chains. Some extensions and examples of double coset Markov chains withGa compact group are discussed.

Homoclinic orbits, multiplier spectrum and rigidity theorems in complex dynamics

Abstract The aims of this paper are to answer several conjectures and questions about the multiplier spectrum of rational maps and giving new proofs of several rigidity theorems in complex dynamics by combining tools from complex and non-Archimedean dynamics. A remarkable theorem due to McMullen asserts that, aside from the flexible Lattès family, the multiplier spectrum of periodic points determines the conjugacy class of rational maps up to finitely many choices. The proof relies on Thurston’s rigidity theorem for post-critically finite maps, in which Teichmüller theory is an essential tool. We will give a new proof of McMullen’s theorem (and therefore a new proof of Thurston’s theorem) without using quasiconformal maps or Teichmüller theory. We show that, aside from the flexible Lattès family, the length spectrum of periodic points determines the conjugacy class of rational maps up to finitely many choices. This generalizes the aforementioned McMullen’s theorem. We will also prove a rigidity theorem for marked length spectrum. Similar ideas also yield a simple proof of a rigidity theorem due to Zdunik. We show that a rational map is exceptional if and only if one of the following holds: (i) the multipliers of periodic points are contained in the integer ring of an imaginary quadratic field, and (ii) all but finitely many periodic points have the same Lyapunov exponent. This solves two conjectures of Milnor.

E-Polynomials of Generic -Character Varieties: Branched Case

Abstract For any branched double covering of compact Riemann surfaces, we consider the associated character varieties that are unitary in the global sense, which we call $\operatorname {\mathrm {GL}}_n\rtimes \!<\!\sigma {>}$ -character varieties. We restrict the monodromies around the branch points to generic semi-simple conjugacy classes contained in $\operatorname {\mathrm {GL}}_n\sigma $ and compute the E-polynomials of these character varieties using the character table of $\operatorname {\mathrm {GL}}_n(q)\rtimes \!<\!\sigma \!>\!$ . The result is expressed as the inner product of certain symmetric functions associated to the wreath product $(\mathbb {Z}/2\mathbb {Z})^N\rtimes \mathfrak {S}_N$ . We are then led to a conjectural formula for the mixed Hodge polynomial, which involves (modified) Macdonald polynomials and wreath Macdonald polynomials.