Intersections of conjugacy classes and subgroups of algebraic groups

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

     

Products of conjugacy classes in finite and algebraic simple groups

We prove the Arad–Herzog conjecture for various families of finite simple groups — if A$A$ and B$B$ are nontrivial conjugacy classes, then AB$AB$ is not a conjugacy class. We also prove that if G$G$ is a finite simple group of Lie type and A$A$ and B$B$ are nontrivial conjugacy classes, either both semisimple or both unipotent, then AB$AB$ is not a conjugacy class. We also prove a strong version of the Arad–Herzog conjecture for simple algebraic groups and in particular show that almost always the product of two conjugacy classes in a simple algebraic group consists of infinitely many conjugacy classes. As a consequence we obtain a complete classification of pairs of centralizers in a simple algebraic group which have dense product. A special case of this has been used by Prasad to prove a uniqueness result for Tits systems in quasi-reductive groups. Our final result is a generalization of the Baer–Suzuki theorem for p$p$-elements with p≥5$p\ge 5$.     

On the conjugacy classes in maximal unipotent subgroups of simple algebraic groups

Let G be a simple algebraic group over the algebraically closed field k of characteristic p ≥ 0. Assume p is zero or good for G. Let B be a Borel subgroup of G; we write U for the unipotent radical of B and u for the Lie algebra of U. Using relative Springer isomorphisms} we analyze the adjoint orbits of U in u. In particular, we show that an adjoint orbit of U in u contains a unique so-called minimal representative. In case p > 0, assume G is defined and split over the finite field of p elements F_p. Let q be a power of p and let G(q) be the finite group of F_q-rational points of G. Let F be the Frobenius morphism such that G(q) = G^F. Assume B is F-stable, so that U is also F-stable and U(q) is a Sylow p-subgroup of G(q). We show that the conjugacy classes of U(q) are in correspondence with the F-stable adjoint orbits of U in u. This allows us to deduce results about the conjugacy classes of U(q).     

Complete reducibility and conjugacy classes of tuples in algebraic groups and Lie algebras

Let H be a reductive subgroup of a reductive group G over an algebraically closed field k. We consider the action of H on G ⁿ , the n-fold Cartesian product of G with itself, by simultaneous conjugation. We give a purely algebraic characterization of the closed H-orbits in G ⁿ , generalizing work of Richardson which treats the case H = G. This characterization turns out to be a natural generalization of Serre??s notion of G-complete reducibility. This concept appears to be new, even in characteristic zero. We discuss how to extend some key results on G-complete reducibility in this framework. We also consider some rationality questions.     

On conjugacy classes in metaplectic groups

Let E be a finite dimensional symplectic space over a local field of characteristic zero. We show that for every element in the metaplectic double cover of the symplectic group Sp(E), and are conjugate by an element of GSp(E) with similitude −1.     

Twisted conjugacy classes in nilpotent groups

Let N be a finitely generated nilpotent group. We show that there is an algorithm that for any automorphism φ∈Aut(N) computes its Reidemeister number R(φ). It is proved that any free nilpotent group N_rc of rank r and class c belongs to class R_∞ if any of the following conditions holds: r=2 and c≥4; r=3 and c≥12; r≥4 and c≥2r.     

Finite groups with small conjugacy classes

We study finite groups G in which the number of distinct prime divisors of the length of the conjugacy classes is at most three. In particular we prove, under this condition, a conjecture of B. Huppert on the number of prime divisors of ÷G/Z(G)÷.     

Affine weyl groups and conjugacy classes in Weyl groups

We define a map from an affine Weyl group to the set of conjugacy classes of an ordinary Weyl group. Supported in part by the National Science Foundation.     

Sign conjugacy classes of the alternating groups

A conjugacy class C of a finite group G is a sign conjugacy class if every irreducible character of G takes value 0,1 or ?1 on C. In this paper, we classify the sign conjugacy classes of alternating groups.     

Some finite groups with large conjugacy classes

We derive some properties of a family of finite groups, which was investigated by Camina, Macdonald, and others. For instance, we give information about the Schur multipliers of the class twop-groups in this family. A large part of this paper was written while the author was visiting the Department of Mathematics of the University of Trento. The author is indebted to this department, and in particular to C.M. Scoppola, for their kind hospitality. The author is also grateful to D. Chillag for his constructively destructive criticism of the first version of this paper.     

Finite groups with three rational conjugacy classes

We show that if a finite group G has exactly three rational conjugacy classes, then G also has exactly three rational-valued irreducible complex characters. This generalizes a result of Navarro and Tiep (Trans Amer Math Soc 360:2443–2465, 2008) and partially answers in the affirmative a conjecture of theirs. We also give a family of examples of non-solvable groups with exactly three rational conjugacy classes.     

Twisted conjugacy classes in saturated weakly branch groups

For a wide class of saturated weakly branch groups, including the (first) Grigorchuk group and the Gupta-Sidki group, we prove that the Reidemeister number of any automorphism is infinite.      

Finite groups with many product conjugacy classes

We classify all finite groupsG such that the product of any two non-inverse conjugacy classes ofG is always a conjugacy class ofG. We also classify all finite groupsG for which the product of any twoG-conjugacy classes which are not inverse modulo the center ofG is again a conjugacy class ofG.     

Finite groups have even more conjugacy classes

In his paper Finite groups have many conjugacy classes (J. London Math. Soc (2) 46 (1992), 239–249), L. Pyber proved the to-date best general lower bounds for the number of conjugacy classes of a finite group in terms of the order of the group. In this paper we strengthen the main results in Pyber’s paper.     

Kazhdan constants for conjugacy classes of compact groups

Kazhdan constants relative to conjugacy classes of compact groups are computed. They depend on the nontrivial irreducible characters of the respective group. The result is applied, in particular, to finite groups of Lie type, symmetric groups, and the group SU(n).     

On groups with conjugacy classes of distinct sizes

A finite group G is called an ah-group if any two distinct conjugacy classes of G have distinct cardinality. We show that if G is an ah-group, then the non-abelian socle of G is isomorphic to one of the following:

1. , for 1a5, a≠2.

2. A₈.

3. PSL(3,4)^e, for 1e10.

4. A₅×PSL(3,4)^e, for 1e10.

Based on this result, we virtually show that if G is an ah-group with π(G) 2,3,5,7 , then F(G)≠1, or equivalently, that G has an abelian normal subgroup.In addition, we show that if G is an ah-group of minimal size which is not isomorphic to S₃, then the non-abelian socle of G is either trivial or isomorphic to one of the following:

1. , for 3a5.

2. PSL(3,4)^e, for 1e10.

Our research lead us to interesting results related to transitivity and homogeneousity in permutation groups, and to subgroups of wreath products of form Z₂S_n. These results are of independent interest and are located in appendices for greater autonomy.     

Small conjugacy classes in the automorphism groups of relatively free groups

Let F be an infinitely generated free group and let R be a fully invariant subgroup of F such that (a) R is contained in the commutator subgroup F^′ of F and (b) the quotient group F/R is residually torsion-free nilpotent. Then the automorphism group of the group F/R^′ is complete. In particular, the automorphism group of any infinitely generated free solvable group of derived length at least two is complete.This extends a result by Dyer and Formanek (1977) [7] on finitely generated groups F_n/R^′ where F_n is a free group of finite rank n at least two and R a characteristic subgroup of F_n.     

On the geometry of conjugacy classes in classical groups

Summary We study closures of conjugacy classes in the Lie algebras of the orthogonal and symplectic groups and determine which ones are normal varieties. Furthermore we give a complete classification of the minimal singularities which arise in this context, i.e. the singularities which occur in the open classes in the boundary of a given conjugacy class. In contrast to the results for the general linear group ([KP1], [KP2]) there are classes with non normal closure; they are branched in a class of codimension two and give rise to normal minimal singularities. The methods used are (classical) invariant theory and algebraic geometry. Supported in part by the SFB Theoretische Mathematik, University of Bonn, and by the University of Hamburg