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Motivated in part by representation theoretic questions, we prove that if G is a finite quasi-simple group, then there exists an elementary abelian subgroup of G that contains a member of each conjugacy class of involutions of G.

Let $G$ be a finite group of order divisible by a prime $p$. The number of $p$-regular and $p'$-regular conjugacy classes of $G$ is at least $2\sqrt{p-1}$. Also, the number of $p$-rational and $p'$-rational irreducible characters of $G$ is at least $2\sqrt{p-1}$. Along the way we prove a uniform lower bound for the number of $p$-regular classes in a finite simple group of Lie type in terms of its rank and size of the underlying field., 46 pages

If C is a class of groups, a C -injector of a finite group G is a subgroup V of G with the property that V ∩ K is a C -maximal subgroup of K for all subnormal subgroups K of G. The classical result of B. Fischer, W. Gaschutz and B. Hartley states the existence and conjugacy of F -injectors in finite soluble groups for Fitting classes F . We shall show that for groups of nilpotent length at most 4, F -injectors permute with the members of a Sylow basis in the group. We shall exhibit the construction of a Fitting class and a group of nilpotent length 5, which fail to satisfy the result and show that the bound is the best possible.

In this article, a possible supercharacter theory based on the degrees of the irreducible characters of a finite group is introduced. An investigation is then included into which finite groups allow this particular supercharacter theory. In particular, it is shown that groups with at most seven conjugacy classes must allow this supercharacter theory. However, there are groups with eight conjugacy classes that do not have this supercharacter theory. In addition, it is shown that semidirect products $$A \rtimes B$$ where both A and B are cyclic groups and at least one is of prime order always allow this type of supercharacter theory.

Let G be a finite group and V(G) be the set of all non-central conjugacy classes of G. Consider the conjugacy class graph Γ(G) of G which is defined as follows: its vertex set is the set V(G) and t...

The primary purpose of this paper is to report on the successful enumeration in Magma of representatives of the 195 826 352 conjugacy classes of transitive subgroups of the symmetric group S 48 of degree 48. In addition, we have determined that 25707 of these groups are minimal transitive and that 713 of them are elusive. The minimal transitive examples have been used to enumerate the vertex-transitive groups of degree 48, of which there are 1 538 868 366 , all but 0.1625% of which arise as Cayley graphs. We have also found that the largest number of elements required to generate any of these groups is 10, and we have used this fact to improve previous general bounds of the third author on the number of elements required to generate an arbitrary transitive permutation group of a given degree. The details of the proof of this improved bound will be published as a separate paper.

Let $V$ be a finite-dimensional vector space over the finite field ${\mathbb F}_q$ and suppose $W$ and $\widetilde{W}$ are subspaces of $V$. Two linear transformations $T:W\to V$ and $\widetilde{T}:\widetilde{W}\to V$ are said to be similar if there exists a linear isomorphism $S:V\to V$ with $SW=\widetilde{W}$ such that $S\circ T=\widetilde{T}\circ S $. Given a linear map $T$ defined on a subspace $W$ of $V$, we give an explicit formula for the number of linear maps that are similar to $T$. Our results extend a theorem of Philip Hall that settles the case $W=V$ where the above problem is equivalent to counting the number of square matrices over ${\mathbb F}_q$ in a conjugacy class., 15 pages, 3 figures

Motivated by a 2004 conjecture by the author and J. Sangroniz, Y. Yang has recently proved that if G is solvable then the index in G of the 8th term of the ascending Fitting series is bounded in terms of the largest number of zeros in a row in the character table of G. In this note, we prove this result for arbitrary finite groups and propose a stronger form of the 2004 conjecture. We conclude the paper showing some possible ways to prove this strengthened conjecture.

We show that, up to strong cocycle conjugacy, every countable exact group admits a unique equivariantly $\mathcal {O}_{2}$-absorbing, pointwise outer action on the Cuntz algebra $\mathcal {O}_{2}$ with the quasi-central approximation property (QAP). In particular, we establish the equivariant analogue of the Kirchberg $\mathcal {O}_{2}$-absorption theorem for these groups.

We prove the p-version of a conjecture raised by Isaacs et al. in Proc. AMS [8] on the degree of a primitive character that divides the size of some conjugacy class of a finite solvable group, thus generalizing Marchi's results [10] in J. Algebra 2020.

We prove that all classical affine W-algebras W(g,f), where g is a simple Lie algebra and f is its non-zero nilpotent element, admit an integrable hierarchy of bi-Hamiltonian PDEs, except possibly for one nilpotent conjugacy class in G_2, one in F_4, and five in E_8., Comment: 18 pages

The affine general linear group acting on a vector space over a prime field is a well-understood mathematical object. Its elementary abelian regular subgroups have recently drawn attention in applied mathematics thanks to their use in cryptography as a way to hide or detect weaknesses inside block ciphers. This paper is focused on building a convenient representation of their elements which suits better the purposes of the cryptanalyst. Several combinatorial counting formulas and a classification of their conjugacy classes are given as well., Comment: to appear in Journal of Algebra

We study the regularity of a conjugacy between an Anosov automorphism $L$ of a nilmanifold $N/\Gamma$ and a volume-preserving, $C^1$-small perturbation $f$. We say that $L$ is locally Lyapunov spectrum rigid if this conjugacy is $C^{1+}$ whenever $f$ is $C^{1+}$ and has the same volume Lyapunov spectrum as $L$. For $L$ with simple spectrum, we show that local Lyapunov spectrum rigidity is equivalent to $L$ satisfying both an irreducibility condition and an ordering condition on its Lyapunov exponents., Comment: 40 pages; comments are welcome

The class of Smale regular homeomorphisms of closed topological manifolds, with nonwandering set consisting of a finite number of periodic orbits of hyperbolic type, is considered. This class contains the Morse-Smale diffeomorphisms of smooth closed manifolds. For two Smale regular homomorphisms necessary and sufficient conditions for being conjugate are presented. Bibliography: 26 titles.

We continue the project started in [8] to describe the structure of the finite groups G of characteristic p in terms of their Baumann components and the conjugacy class Bau p ( G ) . The reduction theorem proved in [8] allows to assume that G has a unique Baumann component. In this paper we use this property to determine the isomorphism type of G / O p ( G ) and the action of G on Ω 1 ( Z ( O p ( G ) ) ) . In addition, we prove reduction theorems which allow to focus on groups G which satisfy G / O p ( G ) ≅ SL n ( q ) , Sp 2 n ( q ) or G 2 ( q ) and O p ( G ) ⩽ B for B ∈ Bau p ( G ) .

There is a natural conjugation action on the set of endomorphism of $��^N$ of fixed degree $d \geq 2$. The quotient by this action forms the moduli of degree $d$ endomorphisms of $��^N$, denoted $\mathcal{M}_d^N$. We construct invariant functions on this moduli space coming from to set of multiplier matrices of the periodic points. The basic properties of these functions are demonstrated such as that they are in the ring of regular functions of $\mathcal{M}_d^N$, methods of computing them, as well as the existence of relations. The main part of the article examines to what extend these invariant functions determine the conjugacy class in the moduli space. Several different types of isospectral families are constructed and a generalization of McMullen's theorem on the multiplier mapping of dimension 1 is proposed. Finally, this generalization is shown to hold when restricted to several specific families in $\mathcal{M}_d^N$., 22 pages

In 2008 , Diaconis and Isaacs introduced the notion of a supercharacter theory of a finite group in which supercharacters replace with irreducible characters and superclasses by conjugacy classes. In this paper, we introduce an algorithm for constructing supercharacter theories of a finite group by which all supercharacter theories of groups containing up to 14 conjugacy classes are calculated.

We say that an ultrafilter on an infinite group $G$ is DTC if it determines the topological centre of the semigroup $\beta G$. We prove that DTC ultrafilters do not exist for virtually BFC groups, and do exist for the countable groups that are not virtually FC. In particular, an infinite finitely generated group is virtually abelian if and only if it does not admit a DTC ultrafilter., Comment: 8 pages. Version 2 includes changes in section 3 and a correction in Example 5.4. Version 3 incorporates minor editing changes

Let G be a finite group, p a prime, and IBr p ( G ) the set of irreducible p-Brauer characters of G. Let e ¯ p ( G ) be the largest integer such that p e ¯ p ( G ) divides χ ( 1 ) for some χ ∈ IBr p ( G ) . We show that | G : O p ( G ) | p ≤ p k e ¯ p ( G ) for an explicitly given constant k. We also study the analogous problem for the p-parts of the conjugacy class sizes of p-regular elements of finite groups.

We previously have developed two methods (Key–Moori Methods 1 and 2) for constructing codes and designs from finite groups (mostly simple finite groups). In this article, we introduce a new method ...

This article is a contribution to the following problem: does there exist a Polish non-archimedean group (equivalently: automorphism group of a Fraisse limit) that is extremely amenable, and has ample generics. As Fraisse limits whose automorphism groups are extremely amenable must be ordered, i.e., equipped with a linear ordering, we focus on ordered Fraisse limits. We prove that automorphism groups of the universal ordered boron tree, and the universal ordered poset have a comeager conjugacy class but no comeager 2-dimensional diagonal conjugacy class. We formulate general conditions implying that there is no comeager conjugacy class, comeager 2-dimensional diagonal conjugacy class or non-meager 2-dimensional topological similarity class in the automorphism group of an ordered Fraisse limit. We also provide a number of applications of these results.

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 present whether the exponent 4 can be improved in general, though we note that the exponent 3 suffices for almost simple groups G. We are however able to prove that every finite group $G$ of even order contains an involution u such that [G:F(G)] is less than the cube of the order of the centralizer of u. There is a dichotomy in the proof of this fact, as it reduces to proving two residual cases: one in which G is almost simple (where the classification of the finite simple groups is needed) and one when G has a Sylow 2-subgroup of order 2. For the latter result, the classification of finite simple groups is not needed (though the Feit-Thompson odd order theorem is). We also prove a very general result on fixed point spaces of involutions in finite irreducible linear groups which does not make use of the classification of the finite simple groups, and some other results on the existence of non-central elements (not necessarily involutions) with large centralizers in general finite groups. We also show (without the classification of finite simple groups) that if t is an involution in G and p is a prime divisor of [G:F(G)], then p is at most 1 plus the order of the centralizer of t (and this is best possible)., Comment: Version 2 consists of some reorganization and some new results

A GLd-pseudocharacter is a function from a group Γ to a ring k satisfying polynomial relations that make it “look like” the character of a representation. When k is an algebraically closed field of characteristic 0, Taylor proved that GLd-pseudocharacters of Γ are the same as degree-d characters of Γ with values in k, hence are in bijection with equivalence classes of semisimple representations Γ → GLd(k). Recently, V. Lafforgue generalized this result by showing that, for any connected reductive group H over an algebraically closed field k of characteristic 0 and for any group Γ, there exists an infinite collection of functions and relations which are naturally in bijection with H(k)-conjugacy classes of semisimple homomorphisms Γ→ H(k). In this paper, we reformulate Lafforgue's result in terms of a new algebraic object called an FFG algebra. We then define generating sets and generating relations for these objects and show that, for all H as above, the corresponding FFG-algebra is finitely presented up to radical. Hence one can always define H-pseudocharacters consisting of finitely many functions satisfying finitely many relations. Next, we use invariant theory to give explicit finite presentations up to radical of the FFG-algebras for (general) orthogonal groups, (general) symplectic groups, and special orthogonal groups. Finally, we use our pseudocharacters to answer questions about conjugacy vs. element-conjugacy of homomorphisms, following Larsen.

We derive a lower and an upper bound for the rank of the finite part of operator $K$-theory groups of maximal and reduced $C^*$-algebras of finitely generated groups. The lower bound is based on the amount of polynomially growing conjugacy classes of finite order elements in the group. The upper bound is based on the amount of torsion elements in the group. We use the lower bound to give lower bounds for the structure group $S(M)$ and the group of positive scalar curvature metrics $P(M)$ for an oriented manifold $M$. We define a class of groups called "polynomially full groups" for which the upper bound and the lower bound we derive are the same. We show that the class of polynomially full groups contains all virtually nilpotent groups. As example, we give explicit formulas for the ranks of the finite parts of operator $K$-theory groups for the finitely generated abelian groups, the symmetric groups and the dihedral groups.

Let K = x G {K=x^{G}} be the conjugacy class of an element x of a group G, and suppose K is finite. We study the increasing sequence of natural numbers { | K n | } n ≥ 1 {\{\lvert K^{n}\rvert\}_{n\geq 1}} and consider restrictions on this sequence and the algebraic consequences. In particular, we prove that if | K 2 | < 3 2 ⁢ | K | {\lvert K^{2}\rvert or if | K 4 | < 2 ⁢ | K | {\lvert K^{4}\rvert , then K n {K^{n}} is a coset of the normal subgroup [ x , G ] {[x,G]} for all n ≥ 2 {n\geq 2} or 4, respectively. We then use these results to contribute to conjectures about the solubility of 〈 K 〉 {\langle K\rangle} when K n {K^{n}} satisfies certain conditions.

Let G be a group, and let [Formula: see text](G) be a digraph whose vertices are the non-trivial conjugacy classes of G and there is an arc from a vertex C to a vertex C′ if and only if C ≠ C′ and 〈C〉 ⊆ 〈C′〉. In this paper, we characterize finite groups G whose associated digraphs [Formula: see text](G) are oriented trees.

In this article we study the involutions of O ( V , q ) , an orthogonal group for a vector space V with quadratic form q over a field of characteristic two. The classification proceeds by discussing conjugacy classes of involutions arising as a product of transvections, involutions with respect to a hyperbolic space, and involutions acting nontrivially in the radical of V. We achieve a complete classification of the conjugacy classes of involutions when the quadratic space ( V , q ) is non-defective, and conclude with a discussion of the defective case.

We apply an orbit theorem to a few questions about character degrees. We investigate the relation of the number of conjugacy classes where characters vanish and the length of the solvable groups. A...

Let p be an odd prime and G be a finite split metabelian p-group of exponent p. In this article, we obtain a normal complement of G in V(FG), where F is the field with p elements. Further, assume t...

In this note, we give a criterion for certain HNN extensions of cyclic conjugacy separable (respectively conjugacy separable) groups with infinite cyclic associated subgroups to be again cyclic con...

This is a survey on the recent progress in the theory of finite groups with factorizations and around it, done by the author and his co-authors, and this has no pretensions to cover all topics in this wide area of research. In particular, we only touch the great consequences of the fundamental paper of Liebeck, Praeger and Saxl on maximal factorizations of almost simple finite groups for the theory of groups with factorizations. In each case the reader can find additional references at the end of Section 1. Some of the methods of investigation can be used to obtain information about finite groups in general, nilpotent algebras and related nearrings.

Let G be a finite group, and let cs ( G ) denote the set of sizes of the conjugacy classes of G. The prime graph built on cs ( G ) , that we denote by Δ ( G ) , is the (simple undirected) graph whose vertices are the prime divisors of the numbers in cs ( G ) , and two distinct vertices p, q are adjacent if and only if pq divides some number in cs ( G ) . A rephrasing of the main theorem in [8] is that the complement Δ ‾ ( G ) of the graph Δ ( G ) does not contain any cycle of length 3. In this paper we generalize this result, showing that Δ ‾ ( G ) does not contain any cycle of odd length, i.e., it is a bipartite graph. In other words, the vertex set V ( G ) of Δ ( G ) is covered by two subsets, each inducing a complete subgraph (a clique). As an immediate consequence, setting ω ( G ) to be the maximum size of a clique in Δ ( G ) , the inequality | V ( G ) | ≤ 2 ω ( G ) holds for every finite group G.

In this paper, using a method of construction of \begin{document}$ 1 $\end{document} -designs which are not necessarily symmetric, introduced by Key and Moori in [ 5 ], we determine a number of \begin{document}$ 1 $\end{document} -designs with interesting parameters from the maximal subgroups and the conjugacy classes of the small Ree groups \begin{document}$ ^2G_2(q) $\end{document} . The designs we obtain are invariant under the action of the groups \begin{document}$ ^2G_2(q) $\end{document} .

In the note we show that the conjugacy of points determined by a correlation of a space of pencils is definitionally equivalent to (suitably defined) orthogonality of lines and to a ‘polarity’, introduced in the space in question. This proves that ‘metric geometry of spaces of pencils’ can be equivalently developed as a theory of incidence enlarged by a natural (point $$\leftrightarrow $$ line) or (line $$\leftrightarrow $$ line) relation.

Let G be a finite group. Let further m1,m2,m3 be three positive integers such that m1 and m2 do not divide each other and m1m2 is coprime to m3. We prove that if the set of conjugacy class sizes of...

Assume k is a positive integer and p is an odd prime. Let ν(G) be the number of conjugacy classes of nonnormal subgroups of a finite group G and NCN(p,k)={ν(G)∈[0,kp]∣G is a finite p-group}. The se...

Let A be a finite dimensional algebra over an algebraically closed field with the radical nilpotent of index 2. It is shown that A has finitely many conjugacy classes of left ideals if and only if A is of finite representation type provided that all simple A-modules have dimension at least 6. This is a revised version.

Discrete logarithm problem (DLP) and Conjugacy search problem (CSP) are two important tools for designing public key protocols. However DLP is used over commutative as well as non-commutative platforms but CSP is used only over non-commutative platforms. To harden the security of cryptosystems using DLP and CSP as base problems, various authors have combined these two problems to form a new problem called Discrete logarithm with conjugacy search problem (DLCSP). It has been used to design key exchange protocols and signature schemes over the general linear group with entries from group ring, that is, $$GL_n(\mathbb {F}_q[S_r])$$ . In this paper, we show that, if someone can solve DLP in polynomial time over some finite extension of $$\mathbb {F}_q$$ , then DLCSP over $$GL_n(\mathbb {F}_q[S_r])$$ can also be solved in polynomial time with non-negligible probability.

This paper concerns the enumeration of simultaneous conjugacy classes of [Formula: see text]-tuples of commuting matrices in the upper triangular group [Formula: see text] and unitriangular group [Formula: see text] over the finite field [Formula: see text] of odd characteristic. This is done for [Formula: see text] and [Formula: see text], by computing the branching rules. Further, using the branching matrix thus computed, we explicitly get the commuting probabilities [Formula: see text] for [Formula: see text] in each case.

Let [Formula: see text] be a finite group and [Formula: see text] be the set of its conjugacy class sizes. In the 1980s, Thompson conjectured that the equality [Formula: see text], where [Formula: see text] and [Formula: see text] is simple, implies the isomorphism [Formula: see text]. In a series of papers of different authors, Thompson’s conjecture was proved. In this paper, we show that in some cases it is possible to omit the conditions [Formula: see text] and [Formula: see text] is simple and prove a more general result.

It is still an open problem to determine the conjugacy classes of Borel subalgebras of non-classical type Lie algebras. In this paper, we prove that there are at least 2 conjugacy classes of Borel subalgebras as well as maximal triangulable subalgebras of restricted Cartan type Lie algebras of type W, S and H. We are particularly interested in maximal triangulable subalgebras of [Formula: see text] under some conditions which is called [Formula: see text]-subalgebras (Definition 3.1). We classify the conjugacy classes of [Formula: see text]-subalgebras for [Formula: see text] and determine their representatives. This paper and its sequel [Z. Lin, K. Ou and B. Shu, Geometric Setting of Jacobson–Witt Algebras, preprint] attempt to establish both algebraic and geometric setting for geometric representation theory of [Formula: see text]

The spread of a group $G$, written $s(G)$, is the largest $k$ such that for any nontrivial elements $x_1, \dots, x_k \in G$ there exists $y \in G$ such that $G = \langle x_i, y \rangle$ for all $i$. Burness, Guralnick and Harper recently classified the finite groups $G$ such that $s(G) > 0$, which involved a reduction to almost simple groups. In this paper, we prove an asymptotic result that determines exactly when $s(G_n) \to \infty$ for a sequence of almost simple groups $(G_n)$. We apply probabilistic and geometric ideas, but the key tool is Shintani descent, a technique from the theory of algebraic groups that provides a bijection, the Shintani map, between conjugacy classes of almost simple groups. We provide a self-contained presentation of a general version of Shintani descent, and we prove that the Shintani map preserves information about maximal overgroups. This is suited to further applications. Indeed, we also use it to study $\mu(G)$, the minimal number of maximal overgroups of an element of $G$. We show that if $G$ is almost simple, then $\mu(G) \leqslant 3$ when $G$ has an alternating or sporadic socle, but in general, unlike when $G$ is simple, $\mu(G)$ can be arbitrarily large., Comment: 30 pages; to appear in Journal of Algebra

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 (resp. subalgebra), or a natural generalisation if $G$ is of type $D_n$. From these representatives, it is straightforward to generate representatives for any given class. Along the way we also prove recognition theorems for identifying both the unipotent classes and nilpotent orbits of exceptional algebraic groups., 26 pages

Let G be a finite group and N(G) be its set of conjugacy class sizes. We prove that if L is a finite simple non-abelian group and G is a finite group with trivial center such that N(G)=N(L) then G≃L.

Let F be a finite field of characteristic p > 0. In this article, weobtain a relation between the class length of elements of a finite p-group G inthe normalized unit group V(FG) and its unitary subgroup V*(FG), when pis an odd prime. We also provide the size of the conjugacy class of non-centralelements of a group G in V(FG); where either G is any finite p-group withnilpotency class 2 or G is a p-group with nilpotency class 3 such that |G

We construct a supercharacter theory, and establish the supercharacter table for Sylow p-subgroups G2syl(q) of the Chevalley groups G2(q) of Lie type G2 when p>2. Then we calculate the conjugacy cl...

Let G be a finite group and p a prime. Let cl ⁡ ( G ) {\operatorname{cl}(G)} be the set of conjugacy classes of G, and let ecl p ⁡ ( G ) {\operatorname{ecl}_{p}(G)} be the largest integer such that p ecl p ⁡ ( G ) {p^{\operatorname{ecl}_{p}(G)}} divides | C | {|C|} for some C ∈ cl ⁡ ( G ) {C\in\operatorname{cl}(G)} . We show that if p ≥ 3 {p\geq 3} and ecl p ⁡ ( G ) = 1 {\operatorname{ecl}_{p}(G)=1} , then | G ⁢ : ⁢ O p ⁢ ( G ) | p ≤ p 3 {\lvert G\mskip 1.0mu \mathord{:}\mskip 1.0mu O_{p}(G)\rvert_{p}\leq p^{3}} . This improves the main result of Y. Yang and G. Qian, On p-parts of conjugacy class sizes of finite groups, Bull. Aust. Math. Soc. 97 2018, 3, 406–411.

Let G be a reductive group over a nonperfect field k. We study rationality problems for Serre’s notion of complete reducibility of subgroups of G. In our previous work, we constructed examples of s...

For a finite group G, we define an equivalence relation ∼ on G as follows:∀ x,y∈G x∼y⇔|xG|=|yG|,where xG is the conjugacy class of x in G. The set of sizes of equivalence classes with respect to th...

In this article, we discuss the normal complement problem for metacyclic groups in modular group algebras. If F is the field with p elements and G is a finite split metacyclic p-group of nilpotency...