Non-central Conjugacy Classes Research Articles

A note on conjugacy classes of finite groups

Let G be a finite group and let xG denote the conjugacy class of an element x of G. We classify all finite groups G in the following three cases: (i) Each non-trivial conjugacy class of G together with the identity element 1 is a subgroup of G, (ii) union of any two distinct non-trivial conjugacy classes of G together with 1 is a subgroup of G, and (iii) union of any three distinct non-trivial conjugacy classes of G together with 1 is a subgroup of G.

Boundedly simple groups of automorphisms of trees

A group is boundedly simple if, for some constant N, every nontrivial conjugacy class generates the whole group in N steps. For a large class of trees, Tits proved simplicity of a canonical subgroup of the automorphism group, which is generated by pointwise stabilizers of edges. We prove that only for uniform subdivisions of biregular trees are such groups boundedly simple. In fact these groups are 8-boundedly simple. As a consequence, we prove that if G is boundedly simple and G acts by automorphisms on a tree, then G fixes some vertex of A, or stabilizes some end of A, or the smallest nonempty G-invariant subtree of A is a uniform subdivision of a biregular tree.

On the commuting probability and supersolvability of finite groups

For a finite group $$G$$ , let $$d(G)$$ denote the probability that a randomly chosen pair of elements of $$G$$ commute. We prove that if $$d(G)>1/s$$ for some integer $$s>1$$ and $$G$$ splits over an abelian normal nontrivial subgroup $$N$$ , then $$G$$ has a nontrivial conjugacy class inside $$N$$ of size at most $$s-1$$ . We also extend two results of Barry, MacHale, and Ni She on the commuting probability in connection with supersolvability of finite groups. In particular, we prove that if $$d(G)>5/16$$ then either $$G$$ is supersolvable, or $$G$$ isoclinic to $$A_4$$ , or $$G/\mathbf{Z}(G)$$ is isoclinic to $$A_4$$ .

Groups with finitely many conjugacy classes of non-normal subgroups of infinite rank

It is proved that if a locally soluble group of infinite rank has only finitely many non-trivial conjugacy classes of subgroups of infinite rank, then all its subgroups are normal.

Covering [formula omitted] and products of blockcyclic matrices

Let V be an n-dimensional vector space over a field K (where n∈N,n⩾3,∣K∣⩾4). A linear transformation φ:V→V is called blockcyclic if the minimum rank satisfies δ(φ)⩾n/2. We study products of blockcyclic conjugacy classes in GL(V). Example: If Ω and Φ are products of blockcyclic conjugacy classes in GL(V) then ΩΦ contains a cyclic transformation. The results are used in our study on products of singular similarity classes (Knüppel and Nielsen, in press) [8]. A further application of our methods is the following theorem. Let k∈N⩾n+1, Ω a non-central conjugacy class of GL(V) and π∈GL(V) such that det(π)=det(Ωk). Then π∈Ωk. A similar result was proved in Lev’s article (Lev, 1996) [10].

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 and B are nontrivial conjugacy classes, then AB is not a conjugacy class. We also prove that if G is a finite simple group of Lie type and A and B are nontrivial conjugacy classes, either both semisimple or both unipotent, then 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-elements with p≥5.

Groups with reality and conjugacy conditions

Many results were proved on the structure of finite groups with some‎ ‎restrictions on their real elements and on their conjugacy classes‎. ‎We‎ ‎generalize a few of these to some classes of infinite groups‎. ‎We study groups in which real elements are central‎, ‎groups in which real elements are $2$-elements‎, ‎groups in which all non-trivial classes have the same finite size and $FC$-groups with two non-trivial conjugacy class sizes‎.

On the regularity of a graph related to conjugacy classes of groups

Given a finite group G, denote by Γ(G) the simple undirected graph whose vertices are the (distinct) non-central conjugacy class sizes of G, and for which two vertices of Γ(G) are adjacent if and only if they are not coprime numbers. In this note we prove that Γ(G) is a 2-regular graph if and only if it is a complete graph with three vertices, and Γ(G) is a 3-regular graph if and only if it is a complete graph with four vertices.

Finite groups whose noncentral class sizes have the same p-part for some prime p

We describe the structure of the finite groups in which the sizes of noncentral conjugacy classes have the same p-part, for some prime p. We prove that they are solvable, they have a normal p-complement and their Fitting length is at most three.

CONJUGACY CLASS SIZES OF CERTAIN DIRECT PRODUCTS

AbstractWe 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.

Products of conjugacy classes in simple groups

Let G be a finite group. For a ∈ G, let a G = {a g | g ∈ G} be the conjugacy class of a in G. In this paper, we study a conjecture due to Arad and Herzog which asserts that in a finite non-abelian simple group the product of two nontrivial conjugacy classes is never a single conjugacy class. In particular, we will verify this conjecture for several families of finite simple groups of Lie type.

D-brane Wess-Zumino terms and U-duality

We construct gauge-invariant and U-duality covariant expressions for Wess-Zumino terms corresponding to general Dp-branes (for any p<D) in arbitrary 2<D<11 dimensions. A distinguishing feature of these Wess-Zumino terms is that they contain twice as many scalars as the 10-D compactified dimensions, in line with doubled geometry. We find that for D<10 the charges of the higher-dimensional branes can all be expressed as products of the 0-brane charges, which include the D0-brane and the NS-NS 0-brane charges. We give the general expressions for these charges and show how they determine the non-trivial conjugacy class to which some of the higher-dimensional D-branes belong.

On a Commuting Graph on Conjugacy Classes of Groups

We consider the graph Γ(G), associated with the conjugacy classes of a group G. Its vertices are the nontrivial conjugacy classes of G, and we join two different classes C, D, whenever there exist x ∈ G and y ∈ D such that xy = yx. The aim of this article is twofold. First, we investigate which graphs can occur in various contexts and second, given a graph Γ(G) associated with G, we investigate the possible structure of G. We proved that if G is a periodic solvable group, then Γ(G) has at most two components, each of diameter at most 9. If G is any locally finite group, then Γ(G) has at most 6 components, each of diameter at most 19. Finally, we investigated periodic groups G with Γ(G) satisfying one of the following properties: (i) no edges exist between noncentral conjugacy classes, and (ii) no edges exist between infinite conjugacy classes. In particular, we showed that the only nonabelian groups satisfying (i) are the three finite groups of order 6 and 8.

Formula omitted]-singular dichotomy for orbital measures of classical compact Lie groups

We prove that for any classical, compact, simple, connected Lie group G, the G-invariant orbital measures supported on non-trivial conjugacy classes satisfy a surprising L 2 -singular dichotomy: Either μ h k ∈ L 2 ( G ) or μ h k is singular to the Haar measure on G. The minimum exponent k for which μ h k ∈ L 2 is specified; it depends on Lie properties of the element h ∈ G . As a corollary, we complete the solution to a classical problem – to determine the minimum exponent k such that μ k ∈ L 1 ( G ) for all central, continuous measures μ on G. Our approach to the singularity problem is geometric and involves studying the size of tangent spaces to the products of the conjugacy classes.

Groups with few conjugacy classes of non-normal subgroups

Abstract The structure of groups with finitely many non-normal subgroups is well known. In this paper, groups are investigated with finitely many conjugacy classes of non-normal subgroups with a given property. In particular, it is proved that a locally soluble group with finitely many non-trivial conjugacy classes of non-abelian subgroups has finite commutator subgroup. This result generalizes a theorem by Romalis and Sesekin on groups in which every non-abelian subgroup is normal.

Products of Subgroups Which Are Subgroups

We prove conditions for a product of distinct subgroups of an arbitrary group G to be a subgroup of G. In particular, the normal closure of any A ≤ G is equal to the product of some distinct conjugates of A. As an application of the later result we derive constraints on the size of a nontrivial conjugacy class of a finite non-Abelian simple group.

Elements of minimal breadth in finite p-groups and lie algebras

AbstractLet G be a finite p-group, and let M(G) be the subgroup generated by the non-central conjugacy classes of G of minimal size. We prove that this subgroup has class at most 3. A similar result is noted for nilpotent Lie algebras.

The extended covering number of SL n is n + 1

For a special linear group SL n ( K) ( K a field) we want to find the minimal integer k (extended covering number) such that 1 ∈ C 1 ⋯ C k for arbitrary non-central conjugacy classes C i of SL n ( K). Using methods from Chevalley groups, length-theorems on products of simple mappings and theorems on products of cyclic mappings we find that k = n + 1, provided n and the field are not too small.

On groups of odd order with exactly two non-central conjugacy classes of each size

The non-abelian group of order 21 is the only finite group of odd order with exactly two non-central conjugacy classes of each size.

THE FREQUENCY SPACE OF A FREE GROUP

We analyze the structure of the frequency spaceQ(F) of a nonabelian free group F = F(a1,…,ak) consisting of all shift-invariant Borel probability measures on ∂F and construct a natural action of Out(F) on Q(F). In particular we prove that for any outer automorphism ϕ of F the conjugacy distortion spectrum of ϕ, consisting of all numbers ‖ϕ(w)‖/‖w‖, where w is a nontrivial conjugacy class, is the intersection of ℚ and a closed subinterval of ℝ with rational endpoints.