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Given a finite group G with a normal subgroup N, the simple graph $\Gamma _{\textit {G}}( \textit {N} )$ is a graph whose vertices are of the form $|x^G|$ , where $x\in {N\setminus {Z(G)}}$ and $x^G$ is the G-conjugacy class of N containing the element x. Two vertices $|x^G|$ and $|y^G|$ are adjacent if they are not coprime. We prove that, if $\Gamma _G(N)$ is a connected incomplete regular graph, then $N= P \times {A}$ where P is a p-group, for some prime p, $A\leq {Z(G)}$ and $\textbf {Z}(N)\not = N\cap \textbf {Z}(G)$ .

We present a simple proof of the Chebotarev density theorem for finite morphisms of quasi-projective varieties over finite fields following an idea of Fried and Kosters for function fields. The key idea is to interpret the number of rational points with a given Frobenius conjugacy class as the number of rational points of a twisted variety, which is then bounded by the Lang–Weil estimates.

Let $G$ be a finite group. Let $\operatorname{cl}(G)$ be the set of conjugacy classes of $G$ and let $\operatorname{ecl}_{p}(G)$ be the largest integer such that $p^{\operatorname{ecl}_{p}(G)}$ divides $|C|$ for some $C\in \operatorname{cl}(G)$. We prove the following results. If $\operatorname{ecl}_{p}(G)=1$, then $|G:F(G)|_{p}\leq p^{4}$ if $p\geq 3$. Moreover, if $G$ is solvable, then $|G:F(G)|_{p}\leq p^{2}$.

We show that if a finitely generated group$G$has a nonelementary WPD action on a hyperbolic metric space$X$, then the number of$G$-conjugacy classes of$X$-loxodromic elements of$G$coming from a ball of radius$R$in the Cayley graph of$G$grows exponentially in$R$. As an application we prove that for$N\geq 3$the number of distinct$\text{Out}(F_{N})$-conjugacy classes of fully irreducible elements$\unicode[STIX]{x1D719}$from an$R$-ball in the Cayley graph of$\text{Out}(F_{N})$with$\log \unicode[STIX]{x1D706}(\unicode[STIX]{x1D719})$of the order of$R$grows exponentially in$R$.

Let a prime $p$ divide the order of a finite real reflection group. We classify the reflection subgroups up to conjugacy that are minimal with respect to inclusion, subject to containing a $p$-Sylow subgroup. For Weyl groups, this is achieved by an algorithm inspired by the Borel–de Siebenthal algorithm. The cases where there is not a unique conjugacy class of reflection subgroups minimally containing the $p$-Sylow subgroups are the groups of type $F_{4}$ when $p=2$ and $I_{2}(m)$ when $m\geq 6$ is even but not a power of $2$ for each odd prime divisor $p$ of $m$. The classification significantly reduces the cases required to describe the $p$-Sylow subgroups of finite real reflection groups.

Let G be a finite group and let N be a normal subgroup of G. We determine the structure of N when the diameter of the graph associated to the G-conjugacy classes contained in N is as large as possible, that is, equal to three., The research of the first and the second authors is supported by the Valencian Government, Proyecto PROMETEOII/2015/011. The first and the third authors are also partially supported by Universitat Jaume I, grant P11B2015-77.

Assume that $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}m$ and $n$ are two positive integers which do not divide each other. If the set of conjugacy class sizes of primary and biprimary elements of a group $G$ is $\{1, m, n, mn\}$, we show that up to central factors $G$ is a $\{p,q\}$-group for two distinct primes $p$ and $q$.

The objective of this paper is to find some sufficient conditions to ensure the conjugacy of supplements of a normal subgroup of a soluble group.

A group $G$ is said to be an $FC$-group if each element of $G$ has only finitely many conjugates, and $G$ is minimal non$FC$ if all its proper subgroups have the property $FC$ but $G$ is not an $FC$-group. It is an open question whether there exists a group of infinite rank which is minimal non$FC$. We consider here groups of infinite rank in which all proper subgroups of infinite rank are $FC$, and prove that in most cases such groups are either $FC$-groups or minimal non$FC$.

The unsolved problem of whether there exists a positive constant $c$ such that the number $k(G)$ of conjugacy classes in any finite group $G$ satisfies $k(G) \geq c \log _{2}|G|$ has attracted attention for many years. Deriving bounds on $k(G)$ from (that is, reducing the problem to) lower bounds on $k(N)$ and $k(G/N)$, $N\trianglelefteq G$, plays a critical role. Recently Keller proved the best lower bound known for solvable groups: \[ k(G)\gt c_{0} \frac {\log _{2}|G|} {\log _{2} \log _{2} |G|}\quad (|G|\geq 4) \] using such a reduction. We show that there are many reductions using $k(G/N) \geq \beta [G : N]^{\alpha }$ or $k(G/N) \geq \beta (\log [G : N])^{t}$ which, together with other information about $G$ and $N$ or $k(N)$, yield a logarithmic lower bound on $k(G)$.

We consider finite groups in which, for all primes p, the p-part of the length of any conjugacy class is trivial or fixed. We obtain a full description in the case in which for each prime divisor p of the order of the group there exists a noncentral conjugacy class of p-power size.

Let G be a finite p-solvable group. We prove that if G has exactly two conjugacy class sizes of p'-elements of prime power order, say 1 and m, then m = p(a)q(b), for two distinct primes p and q, and G either has an abelian p-complement or G = PQ x A, with P and Q a Sylow p-subgroup and a Sylow q-subgroup of G, respectively, and A is abelian. In particular, we provide a new extension of Ito's theorem on groups having exactly two class sizes for elements of prime power order., This research is supported by the Spanish Government, Proyecto MTM2010-19938-C03-02 and the second and third authors are supported by the Valencian Government, Proyecto PROMETEO/2011/30. The second author is also supported by grant Fundacio Caixa-Castello P11B2010-47.

Fix a prime number p. Let G be a p-modular Frobenius group with kernel N which is the minimal normal subgroup of G. We give the complete classification of G when N has three, four or five p-regular conjugacy classes. We also determine the structure of G when N has more than five p-regular conjugacy classes.

We extend some results known for FC-groups to the class FC* of generalized FC-groups introduced in de Giovanni et al. [‘Groups with restricted conjugacy classes’, Serdica Math. J.28(3) (2002), 241–254]. The main theorems pertain to the join of pronormal subgroups. The relevant role that the Wielandt subgroup plays in an FC*-group is pointed out.

We consider finite p-groups G in which every cyclic subgroup has at most p conjugates. We show that the derived subgroup of such a group has order at most p2. Further, if the stronger condition holds that all subgroups have at most p conjugates then the central factor group has order p4 at most.

We compute commutativity degrees of wreath products $A \wr B$ of finite Abelian groups A and B. When B is fixed of order n the asymptotic commutativity degree of such wreath products is 1/n2. This answers a generalized version of a question posed by P. Lescot. As byproducts of our formula we compute the number of conjugacy classes in such wreath products, and obtain an interesting elementary number-theoretic result.

Let G be a finite group and π an arbitrary set of primes. We investigate the structure of G when the lengths of the conjugacy classes of its π-elements are prime powers. Under this condition, we show that such lengths are either powers of just one prime or exactly {1,qa, rb}, with q and r two distinct primes lying in π and a, b > 0. In the first case, we obtain certain properties of the normal structure of G, and in the second one, we provide a characterisation of the structure of G.

We show the existence of a convex representation of a maximal monotone operator by a convex function which is invariant with respect to the Fenchel conjugacy (up to an interchange of variables). We use the framework of generalized convexity.

Let p be an odd prime and let G be a finite p-group such that xZ(G)⊆xG, for all x∈G∖Z(G), where xG denotes the conjugacy class of x in G. Then G has a noninner automorphism of order p leaving the Frattini subgroup Φ(G) elementwise fixed.

OUTLINE. In the first half of the thesis, we introduce the class of'chainable monoids' and describe general techniques for solving the word and division problems and for obtaining normal forms for monoids in this class. Their growth functions are shown to be rational, and calculable. We also show that a monoid with a certain central 'fundamental element' embeds in a group or related monoid. Motivating examples of monoids in this class are positive braid monoids and positive singular braid monoids. Much of the rest of the thesis is devoted to applying the techniques described to singular Artin monoids, and we obtain further results regarding conjugacy and parabolic submonoids. In the final chapter we apply the techniques of chainable monoids to braid groups of complex reflection groups. A chainable monoid is a monoid which may be defined by a finite presentation with four properties; three of which can be determined immediately from the length of the relations and their starting letters, and the fourth, the reduction property, which is more involved to verify. It has been shown that positive Artin monoids, including braid monoids, have these properties. Other examples of monoids in this class include free monoids, free commutative monoids, positive singular braid monoids (see later), a large class of one relation monoids, trace monoids and divisibility monoids. The class of chainable monoids is closed under taking direct and free products. Chainable monoids are always left cancellative. The method of chains enables the construction of a calculable partial map on pairs of words, which returns the left quotient of the first word by the second, whenever it exists, thus giving a solution to the division problem, and hence word problem, in a chainable monoid. Again using chains, we construct another partial map on pairs of words which is shown to return their least common left multiple, whenever it exists. We show that in a chainable monoid, greatest common left divisors always exist, and least common left multiples exist whenever common left multiples exist. [The author is grateful to Dietrich Kuske for recently pointing out that chainable monoids may be algebraically characterised as left cancellative monoids that

Let G be a finite group and let p be a prime number. We consider the Submatrix of the character table of G whose rows are indexed by the characters in blocks of maximal defect, and whose columns are indexed by the conjugacy classes of P′-size. We prove that this matrix has maximum rank.

Every group acts transitively by conjugation on each of its conjugacy classes of elements. It is natural to wonder when this action becomes multiply transitive. In this paper, we determine all finite groups which act faithfully and 2-transitively on a conjugacy class of elements. We also give some consequences including a solvability criterion based on what fraction of elements belong to conjugacy classes upon which the group acts faithfully and 2–transitively.

If G is a finite p-group of order pn, P. Hall determined the number of conjugacy classes of G, r(G), modulo (p2 − 1)(p − 1). Namely, he proved the existence of a constant k ≥ 0 such that r(G) = n(p2 − 1) + pe + k(p2 − 1)(p − 1). In this paper, we denote by the group of the upper unitriangular matrices over , the finite field with q = pt elements, and we determine the number of classes of modulo (q − 1)5.

A conjugacy operation is introduced on the set Q(X) of all quasiconvex lower semicontinuous nonnegative functions vanishing at zero. This operation is used in order to introduce and study a dual problem with respect to a maximisation problem where both constraint and objective functions belong to Q(X).

Let $G$ be a finite $p$-solvable group and let ${G}^{\ast } $ be the set of elements of primary and biprimary orders of $G$. Suppose that the conjugacy class sizes of ${G}^{\ast } $ are $\{ 1, {p}^{a} , n, {p}^{a} n\} $, where the prime $p$ divides the positive integer $n$ and ${p}^{a} $ does not divide $n$. Then $G$ is, up to central factors, a $\{ p, q\} $-group with $p$ and $q$ two distinct primes. In particular, $G$ is solvable.

Let G be a Weyl group of type B, and T a set of representatives of the conjugacy classes of self-inverse elements of G. For each t in T, we construct a (complex) linear character πt of the centraliser of t in G, such that the sum of the characters of G induced from the πt contains each irreducible complex character of G with multiplicity precisely 1. For Weyl groups of type A (that is, for the symmetric groups), a similar result was published recently by Inglis, Richardson and Saxl.

The conjugacy classes in the finite-dimensional projective full linear, special linear and projective special linear groups over an arbitrary commutative field are determined. The results over a finite field are applied to certain enumerative problems.

Laue et al have described basic algorithms for computing in a finite soluble group G given by an AG-presentation, among them a general algorithm for the computation of the orbits of such a group acting on some set Ω. Among other applications, this algorithm yields straightforwardly a method for the computation of the conjugacy classes of elements in such a group, which has been implemented in 1986 in FORTRAN within SOGOS by the first author and in 1987 in C within CAYLEY. However, for this particular problem one can do better, as discussed in this note.

The rôle of the conjugacy relation is certainly important in the structure theory of groups. Here we study this relation in a considerably more general setting, namely in the theory of loops. We first recall some basic facts about quasigroups, their multiplication groups, their inner mapping groups and the conjugacy relation. After this we estimate the size and the number of the conjugacy classes and we study the structure of loops having only two conjugacy classes. Finally, the values of the centraliser function are discussed.

In this paper we give an elementary method for classifying conjugacy classes of involutions in a Coxeter group (W, S). The classification is in terms of (W-equivalence classes of certain subsets of S).

A duality theory for finite abelian hypergroups over fairly general fields is presented, which extends the classical duality for finite abelian groups. In this precise sense the set of conjugacy classes and the set of characters of a finite group are dual as hypergroups.

In this paper we calculate the number of congugacy classes in the following finite classical groups: GLn(Fq); PGLn(Fq), SLn(Fq), and more generally G(Fq), where G is any algebraic group isogenous to SLn; PSLn(Fq); ; , , and more generally where G is any group isogenous to SUn over Fq; and .

The category of groups forms a full subcategory of the category of hypergroups. This larger category also contains other group theoretic objects, such as the conjugacy class hypergroup and character hypergroup of a finite group.Definitions of hypergroups and hypergroup morphisms are given and related to double algebras (which are simultaneously algebras and cogebras, but not Hopf algebras in general). Quotient and orbit hypergroups are defined. Coefficients are allowed in a more or less arbitrary field.These concepts provide a new language in which groups and their character tables can be fruitfully discussed.

We give some results on the number of G-orbits (G = GL(n, q)) on groups isogenous (in the algebraic-geometric sense) to SL(n, q). The conjectured isogeny invariance of this number is contradicted by computer calculations.

It is shown that there exists a logarithmic upper bound on the Fitting length h(G) of a finite soluble group G in terms of the number ν(G) of the conjugacy classes of its maximal nilpotent subgroups. For ν(G) = 3, the best possible bound on h(G) is shown to be 4.

In a finite soluble group G, the Fitting (or nilpotency) length h(G) can be considered as a measure for how strongly G deviates from being nilpotent. As another measure for this, the number v(G) of conjugacy classes of the maximal nilpotent subgroups of G may be taken. It is shown that there exists an integer-valued function f on the set of positive integers such that h(G) ≦ f(v(G)) for all finite (soluble) groups of odd order. Moreover, if all prime divisors of the order of G are greater than v(G)(v(G) - l)/2, then h(G) ≦3. The bound f(v(G)) is just of qualitative nature and by far not best possible. For v(G) = 2, h(G) = 3, some statements are made about the structure of G.