In earlier work, the first author outlined an algorithm for calculating a parametrization of the conjugacy classes in a Sylow $p$-subgroup $U(q)$ of a finite Chevalley group $G(q)$, valid when $q$ is a power of a good prime for $G(q)$. In this paper we develop this algorithm and discuss an implementation in the computer algebra language {\sf GAP}. Using the resulting computer program we are able to calculate the parametrization of the conjugacy classes in $U(q)$, when $G(q)$ is of rank at most 6. In these cases, we observe that the number of conjugacy classes of $U(q)$ is given by a polynomial in $q$ with integer coefficients.
Let $U$ be a Sylow $p$-subgroup of the finite Chevalley group of type $D_4$ over the field of $q$ elements, where $q$ is a power of a prime $p$. We describe a construction of the generic character table of $U$.
We develop in this work a method to parametrize the set $\mathrm{Irr}(U)$ of irreducible characters of a Sylow $p$-subgroup $U$ of a finite Chevalley group $G(p^f)$ which is valid for arbitrary primes $p$, in particular when $p$ is a very bad prime for $G$. As an application, we parametrize $\mathrm{Irr}(U)$ when $G=\mathrm{F}_4(2^f)$.
Let $U(q)$ be a Sylow $p$-subgroup of the Chevalley groups $D_4(q)$ where $q$ is a power of a prime $p$. We describe a construction of all complex irreducible characters of $U(q)$ and obtain a classification of these irreducible characters via the root subgroups which are contained in the center of these characters. Furthermore, we show that the multiplicities of the degrees of these irreducible characters are given by polynomials in $q-1$ with nonnegative coefficients.
Let $\F_q$ be a field of characteristic $p$ with $q$ elements. It is known that the degrees of the irreducible characters of the Sylow $p$-subgroup of $GL_n(\F_q)$ are powers of $q$ by Issacs. On the other hand Sangroniz showed that this is true for a Sylow $p$-subgroup of a classical group defined over $\F_q$ if and only if $p$ is odd. For the classical groups of Lie type $B$, $C$ and $D$ the only bad prime is 2. For the exceptional groups there are others. In this paper we construct irreducible characters for the Sylow $p$-subgroups of the Chevalley groups $D_4(q)$ with $q=2^f$ of degree $q^…
Let $q$ be a power of a prime $p$, let $G$ be a finite Chevalley group over $\mathbb{F}_q$ and let $U$ be a Sylow $p$-subgroup of $G$; we assume that $p$ is not a very bad prime for $G$. We explain a procedure of reduction of irreducible complex characters of $U$, which leads to an algorithm whose goal is to obtain a parametrization of the irreducible characters of $U$ along with a means to construct these characters as induced characters. A focus in this paper is determining the parametrization when $G$ is of type $\mathrm{F}_4$, where we observe that the parametrization is "uniform" over goo…
We parametrize the set of irreducible characters of the Sylow $p$-subgroups of the Chevalley groups $\mathrm{D}_6(q)$ and $\mathrm{E}_6(q)$, for an arbitrary power $q$ of any prime $p$. In particular, we establish that the parametrization is uniform for $p \ge 3$ in type $\mathrm{D}_6$ and for $p \ge 5$ in type $\mathrm{E}_6$, while the prime $2$ in type $\mathrm{D}_6$ and the primes $2,$ $3$ in type $\mathrm{E}_6$ yield character degrees of the form $q^m/p^i$ which force a departure from the generic situations. Also for the first time in our analysis we see a family of irreducible characters …
Let $q$ be a power of a prime $p$ and let $U(q)$ be a Sylow $p$-subgroup of a finite Chevalley group $G(q)$ defined over the field with $q$ elements. We first give a parametrization of the set $\text{Irr}(U(q))$ of irreducible characters of $U(q)$ when $G(q)$ is of type $\mathrm{G}_2$. This is uniform for primes $p \ge 5$, while the bad primes $p=2$ and $p=3$ have to be considered separately. We then use this result and the contribution of several authors to show a general result, namely that if $G(q)$ is any finite Chevalley group with $p$ a bad prime, then there exists a character $χ\in \tex…
Let $U = \mathbf U(q)$ be a Sylow $p$-subgroup of a finite Chevalley group $G = \mathbf G(q)$. In [GR}] Röhrle and the second author determined a parameterization of the conjugacy classes of $U$, for $\mathbf G$ of small rank when $q$ is a power of a good prime for $\mathbf G$. As a consequence they verified that the number $k(U)$ of conjugacy classes of $U$ is given by a polynomial in $q$ with integer coefficients. In the present paper, we consider the case when $p$ is a bad prime for $\mathbf G$. We obtain a parameterization of the conjugacy classes of $U$, when $\mathbf G$ has rank less tha…
We prove that Chevalley group over the field $F$ of zero characteristic possess $R_{\infty}$ property, if $F$ has torsion group of automorphisms or $F$ is an algebraically closed field which has finite transcendence degree over $\mathbb{Q}$. As a consequence we obtain that the twisted conjugacy class $[e]_φ$ of unit element is a subgroup of Chevalley group if and only if $φ$ is central automorphism.
Let G(q) be a finite Chevalley group, where q is a power of a good prime p, and let U(q) be a Sylow p-subgroup of G(q). Then a generalized version of a conjecture of Higman asserts that the number k(U(q)) of conjugacy classes in U(q) is given by a polynomial in q with integer coefficients. In an earlier paper, the first and the third authors developed an algorithm to calculate the values of k(U(q)). By implementing it into a computer program using GAP, they were able to calculate k(U(q)) for G of rank at most 5, thereby proving that for these cases k(U(q)) is given by a polynomial in q. In thi…
We present explicit upper bounds for the number and size of conjugacy classes in finite Chevalley groups and their variations. These results have been used by many authors to study zeta functions associated to representations of finite simple groups, random walks on Chevalley groups, the final solution to the Ore conjecture about commutators in finite simple groups and other similar problems. In this paper, we solve a strong version of the Boston-Shalev conjecture on derangements in simple groups for most of the families of primitive permutation group representations of finite simple groups (t…
Let $\ell$ be a prime divisor of the order of a finite unitary reflection group. We classify up to conjugacy the parabolic and reflection subgroups that are minimal with respect to inclusion, subject to containing an $\ell$-Sylow subgroup. The classification assists in describing the $\ell$-Sylow subgroups of unitary reflection groups up to group isomorphism. This classification also relates to the modular representation theory of finite groups of Lie type. We observe that unless a parabolic subgroup minimally containing an $\ell$-Sylow subgroup is $G$ itself, the reflection subgroup within th…
Broué, Malle and Michel have shown that the automizer of an abelian Sylow p-subgroup in a finite simple Chevalley group is an irreducible complex reflection group, for p not too small and different from the defining characteristic. The aim is this note is to show that a suitable version of this property holds for general finite groups. As an example, the automizer of an 11-Sylow subgroup in the Monster is the 2-dimensional complex reflection group G_{16}.
The paper studies lower bounds for the dimensions of projective indecomposable modules for Chevalley groups G in defining characteristic p. The main result extending earlier one by Malle and Weigel (2008) determines the modules in question of dimension equal to the order of a Sylow p-subgroup of G. We also substantially generalize a result by Ballard (1978) on lower bounds for the dimensions of projective indecomposable modules and find lower bounds in some cases where Ballard's bounds are vacuous.
Let $p$ be a prime and let $P$ be a Sylow $p$-subgroup of a finite nonabelian group $G$. Let $bcl(G)$ be the size of the largest conjugacy class of the group $G$. We show that $|P/O_p(G)| < bcl(G)$ if $G$ is not abelian.
We express integrals of definable functions over definable sets uniformly for non-Archimedean local fields, extending results of Pas. We apply this to Chevalley groups, in particular proving that zeta functions counting conjugacy classes in congruence quotients of such groups depend only on the size of the residue field, for sufficiently large residue characteristic. In particular, the number of conjugacy classes in a congruence quotient depends only on the size of the residue field. The same holds for zeta functions counting dimensions of Hecke modules of intertwining operators associated to …
Define a random variable $ξ_n$ by choosing a conjugacy class $C$ of the Sylow $p$-subgroup of $S_{p^n}$ by random, and let $ξ_n$ be the logarithm of the order of an element in $C$. We show that $ξ_n$ has bounded variance and mean order $\frac{\log n}{\log p}+O(1)$, which differs significantly from the average order of elements chosen with equal probability.
Let $G$ be a finite group with Sylow subgroups $P_1,\ldots,P_n$, and let $k(G)$ denote the number of conjugacy classes of $G$. Pyber asked if $k(G) \leq \prod_{i=1}^n k(P_i)$ for all finite groups $G$. With the help of GAP, we prove that Pyber's inequality holds for all symmetric and alternating groups.
It is proved that a profinite group $G$ has fewer than $2^{\aleph_0}$ conjugacy classes of $p$-elements for an odd prime $p$ if and only if its $p$-Sylow subgroups are finite. (Here, by a $p$-element one understands an element that either has $p$-power order or topologically generates a group isomorphic to ${\mathbb Z}_p$.) A weaker result is proved for $p=2$.
Let $F$ be a subfield of the algebraic closure of a finite field $\mathbb{F}_p$, $p \ne 2$, and let $R$ denote any ring such that $F[t] \subset R \subsetneq F(t)$. Let $G$ be a classical Chevalley group of adjoint type defined over $R$. We prove that the group $G(R)$ has the $R_{\infty}$-property.
In this paper, we study the structures of finite groups using some arithmetic conditions on the sizes of real conjugacy classes. We prove that a finite group is solvable if the prime graph on the real class sizes of the group is disconnected. Moreover, we show that if the sizes of all non-central real conjugacy classes of a finite group $G$ have the same $2$-part and the Sylow $2$-subgroup of $G$ satisfies certain condition, then $G$ is solvable.
The goal of this investigation is effective method of key exchange which based on non-commutative group $G$. The results of Ko et al. \cite{kolee} is improved and generalized. The size of a minimal generating set for the commutator subgroup of Sylow 2-subgroups of alternating group is found. The structure of the commutator subgroup of Sylow 2-subgroups of the alternating group ${A_{2^{k}}}$ is investigated and used in key exchange protocol which based on non-commutative group. We consider non-commutative generalization of CDH problem \cite{gu2013new, bohli2006towards} on base of metacyclic gro…
Universidad Nacional de Córdoba（科尔多瓦国立大学）；CIEM-CONICET（数学与计算研究所-阿根廷国家科学研究委员会）；Vrije Universiteit Brussel（布鲁塞尔自由大学）；Southern University of Science and Technology（南方科技大学）；Università degli Studi di Padova（帕多瓦大学）
This survey on the automorphism groups of finite p-groups focuses on three major topics: explicit computations for familiar finite p-groups, such as the extraspecial p-groups and Sylow p-subgroups of Chevalley groups; constructing p-groups with specified automorphism groups; and the discovery of finite p-groups whose automorphism groups are or are not p-groups themselves. The material is presented with varying levels of detail, with some of the examples given in complete detail.
Let $G:={^2G_2}(q)$ be the simple Ree group with $q=3^{2k+1}$ and $k$ a positive integer. We show that the centre of the principal block $Z(kGe_0)$, where $k$ is an algebraically closed field of characteristic $3$, is not isomorphic to the centre of the Brauer corresponding block $Z(kN_G(P))$, where $N_G(P)$ is the normaliser in $G$ of a Sylow $3$-subgroup. As part of the proof, we compute the conjugacy classes of elements and the character tables of the maximal parabolic subgroups of $G$.
The conjugacy of split Cartan subalgebras in the finite dimensional simple case (Chevalley) and in the symmetrizable Kac-Moody case (Peterson-Kac) are fundamental results of the theory of Lie algebras. Among the Kac-Moody Lie algebras the affine algebras stand out. This paper deals with the problem of conjugacy for a class of algebras --extended affine Lie algebras-- that are in a precise sense higher nullity analogues of the affine algebras. Unlike the methods used by Peterson-Kac, our approach is entirely cohomological and geometric. It is deeply rooted on the theory of reductive group schem…
Let $p$ be an odd prime and let $B$ be a $p$-block of a finite group which has cyclic defect groups. We show that all exceptional characters in $B$ have the same Frobenius-Schur indicators. Moreover the common indicator can be computed, using the canonical character of $B$. We also investigate the Frobenius-Schur indicators of the non-exceptional characters in $B$. For a finite group which has cyclic Sylow $p$-subgroups, we show that the number of irreducible characters with Frobenius-Schur indicator $-1$ is greater than or equal to the number of conjugacy classes of weakly real $p$-elements i…
Let $q=2^f$, and let $G=\mathrm{SO}_8^+(q)$ and $U$ be a Sylow $2$-subgroup of $G$. We first describe the fusion of the conjugacy classes of $U$ in $G$. We then use this information to prove the unitriangularity of the $\ell$-decomposition matrices of $G$ for all $\ell \ne 2$ by inducing certain irreducible characters of $U$ to $G$; the characters of $U$ of degree $q^3/2$ play here a major role. We then determine the $\ell$-decomposition matrix of $G$ in the case $\ell \mid q+1$, when $\ell \ge 5$ and $(q+1)_\ell>5$, up to two non-negative indeterminates in one column.
We show that if $p$ is a prime and $G$ is a finite $p$-solvable group satisfying the condition that a prime $q$ divides the degree of no irreducible $p$-Brauer character of $G$, then the normalizer of some Sylow $q$-subgroup of $G$ meets all the conjugacy classes of $p$-regular elements of $G$.
We give a classification of the $p$--local stable homotopy type of $BG$, where $G$ is a finite group, in purely algebraic terms. $BG$ is determined by conjugacy classes of homomorphisms from $p$--groups into $G$. This classification greatly simplifies if $G$ has a normal Sylow $p$--subgroup; the stable homotopy types then depends only on the Weyl group of the Sylow $p$--subgroup. If $G$ is cyclic mod $p$ then $BG$ determines $G$ up to isomorphism. The last class of groups is important because in an appropriate Grothendieck group $BG$ can be written as a unique linear combination of $BH$'s, whe…
Let $G$ be a finite group, and let cs$(G)$ be the set of conjugacy class sizes of $G$. Recalling that an element $g$ of $G$ is called a \emph{vanishing element} if there exists an irreducible character of $G$ taking the value $0$ on $g$, we consider one particular subset of cs$(G)$, namely, the set vcs$(G)$ whose elements are the conjugacy class sizes of the vanishing elements of $G$. Motivated by the results in \cite{BLP}, we describe the class of the finite groups $G$ such that vcs$(G)$ consists of a single element \emph{under the assumption that $G$ is supersolvable or $G$ has a normal Sylo…
We prove that a 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 a knowledge of the conjugacy class sizes was known. We prove variations of both results and discuss their similarities.
A group is called strongly bounded, if the speed with which it is generated by finitely many conjugacy classes has a positive, lower bound only dependent on the number of the conjugacy classes in question rather than the actual conjugacy classes. Earlier papers by Kedra, Libman and Martin and myself have shown that this is a property common to split Chevalley groups defined using an irreducible root system of rank at least $2$ and the ring of all S-algebraic integers and that the situation is dependent on the number theory of $R$ for ${\rm Sp}_4$ and $G_2.$ In this paper, we will show that ${\…
Let $G$ be a simple algebraic group over an algebraically closed field $k$ of characteristic $p$. The classification of the conjugacy classes of unipotent elements of $G(k)$ and nilpotent orbits of $G$ on $\operatorname{Lie}(G)$ is well-established. One knows there are representatives of every unipotent class as a product of root group elements and every nilpotent orbit as a sum of root elements. We give explicit representatives in terms of a Chevalley basis for the eminent classes. A unipotent (resp. nilpotent) element is said to be eminent if it is not contained in any subsystem subgroup (re…
Let $N$ be a normal subgroup of a finite group $G$. In this paper, we consider the elements $g$ of $N$ such that $χ(g)\neq 0$ for all irreducible characters $χ$ of $G$. Such an element is said to be non-vanishing in $G$. Let $p$ be a prime. If all $p$-elements of $N$ satisfy the previous property, then we prove that $N$ has a normal Sylow $p$-subgroup. As a consequence, we also study certain arithmetical properties of the $G$-conjugacy class sizes of the elements of $N$ which are zeros of some irreducible character of $G$. In particular, if $N=G$, then new contributions are obtained.
A p-local finite group consists of a finite p-group S, together with a pair of categories which encode ``conjugacy'' relations among subgroups of S, and which are modelled on the fusion in a Sylow p-subgroup of a finite group. It contains enough information to define a classifying space which has many of the same properties as p-completed classifying spaces of finite groups. In this paper, we study and classify extensions of p-local finite groups, and also compute the fundamental group of the classifying space of a p-local finite group.
Let $G$ be a linear algebraic group over an algebraically closed field of characteristic $p\geq 0$. We show that if $H_1$ and $H_2$ are connected subgroups of $G$ such that $H_1$ and $H_2$ have a common maximal unipotent subgroup and $H_1/R_u(H_1)$ and $H_2/R_u(H_2)$ are semisimple, then $H_1$ and $H_2$ are $G$-conjugate. Moreover, we show that if $H$ is a semisimple linear algebraic group with maximal unipotent subgroup $U$ then for any algebraic group homomorphism $σ\colon U\rightarrow G$, there are only finitely many $G$-conjugacy classes of algebraic group homomorphisms $ρ\colon H\rightarr…
We show that the abelian monoid of isomorphism classes of G-stable finite S-sets is free for a finite group G with Sylow p-subgroup S; here a finite S-set is called G-stable if it has isomorphic restrictions to G-conjugate subgroups of S. These G-stable S-sets are of interest, e.g., in homotopy theory. We prove freeness by constructing an explicit (but somewhat non-obvious) basis, whose elements are in one-to-one correspondence with the G-conjugacy classes of subgroups in S. As a central tool of independent interest, we give a detailed description of the embedding of the Burnside ring for a sa…
The aim of this article is to explore global and local properties of finite groups whose integral group rings have only trivial central units, so-called cut groups. For such a group we study actions of Galois groups on its character table and show that the natural actions on the rows and columns are essentially the same, in particular the number of rational-valued irreducible characters coincides with the number of rational-valued conjugacy classes. Further, we prove a natural criterion for nilpotent groups of class 2 to be cut and give a complete list of simple cut groups. Also, the impact of…
A p-local finite group consists of a finite p-group S, together with a pair of categories which encode ``conjugacy'' relations among subgroups of S, and which are modelled on the fusion in a Sylow p-subgroup of a finite group. It contains enough information to define a classifying space which has many of the same properties as p-completed classifying spaces of finite groups. In this paper, we examine which subgroups control this structure. More precisely, we prove that the question of whether an abstract fusion system F over a finite p-group S is saturated can be determined by just looking at …
Brauer and Fowler noted restrictions on the structure of a finite group G in terms of the order of the centralizer of an involution t in G. We consider variants of these themes. We first note that for an arbitrary finite group G of even order, we have |G| is less than the number of conjugacy classes of the Fitting subgroup times the order of the centralizer to the fourth power of any involution in G. This result does require the classification of the finite simple groups. The groups SL(2,q) with q even shows that the exponent 4 cannot be replaced by any exponent less than 3. We do not know at …
