The sub-class sizes of some elements being square free

Let [Formula: see text] be an element of a finite group [Formula: see text], and [Formula: see text] a prime factor of the order of [Formula: see text]. It is clear that there always exists a unique minimal subnormal subgroup containing [Formula: see text], say [Formula: see text]. We call the conjugacy class of [Formula: see text] in [Formula: see text] the sub-class of [Formula: see text] in [Formula: see text], see [G. Qian and Y. Yang, On sub-class sizes of finite groups, J. Aust. Math. Soc. (2020) 402–411]. In this paper, assume that [Formula: see text] is the product of the subgroups [Formula: see text] and [Formula: see text], we investigate the solvability, [Formula: see text]-nilpotence and supersolvability of the group [Formula: see text] under the condition that the sub-class sizes of prime power order elements in [Formula: see text] are [Formula: see text] free, [Formula: see text] free and square free, respectively, so that some known results relevant to conjugacy class sizes are generalized.

Topological conjugacy of 𝑛-multiple Cartesian products of circle rough transformations

It is known from the 1939 work of A. G. Mayer that rough transformations of the circle are limited to the diffeomorphisms of Morse – Smale. A topological conjugacy class of orientation-preserving diffeomorphism is entirely determined by its rotation number and the number of its periodic orbits, while for orientation-changing diffeomorphism the topological invariant will be only the number of periodic orbits. Thus, the purpose of this study is to find topological invariants of n-fold Cartesian products of diffeomorphisms of a circle. Methods. This paper explores the rough Morse – Smale diffeomorphisms on the n-torus surface. To prove the main result, additional constructions and formation of subsets of considered sets were used. Results. In this paper, a numerical topological invariant is introduced for n-fold Cartesian products of rough circle transformations. Conclusion.The criterion of topological conjugacy of n-fold Cartesian products of rough transformations of a circle is formulated.

Remarks on weak amalgamation and large conjugacy classes in non-archimedean groups

We study the notion of weak amalgamation in the context of diagonal conjugacy classes. Generalizing results of Kechris and Rosendal, we prove that for every countable structure M, Polish group G of permutations of M, and $$n \ge 1$$ n ≥ 1 , G has a comeager n-diagonal conjugacy class iff the family of all n-tuples of G-extendable bijections between finitely generated substructures of M, has the joint embedding property and the weak amalgamation property. We characterize limits of weak Fraïssé classes that are not homogenizable. Finally, we investigate 1- and 2-diagonal conjugacy classes in groups of ball-preserving bijections of certain ordered ultrametric spaces.

Non-abelian anyons and some cousins of the Arad-Herzog conjecture

Long ago, Arad and Herzog (AH) conjectured that, in finite simple groups, the product of two conjugacy classes of length greater than one is never a single conjugacy class. We discuss implications of this conjecture for non-abelian anyons in 2 + 1-dimensional discrete gauge theories. Thinking in this way also suggests closely related statements about finite simple groups and their associated discrete gauge theories. We prove these statements and provide some physical intuition for their validity. Finally, we explain that the lack of certain dualities in theories with non-abelian finite simple gauge groups provides a non-trivial check of the AH conjecture.

Classifying finite monomial linear groups of prime degree in characteristic zero

Let [Formula: see text] be a prime and let [Formula: see text] be the complex field. We explicitly classify the finite solvable irreducible monomial subgroups of [Formula: see text] up to conjugacy. That is, we give a complete and irredundant list of [Formula: see text]-conjugacy class representatives as generating sets of monomial matrices. Copious structural information about non-solvable finite irreducible monomial subgroups of [Formula: see text] is also proved, enabling a classification of all such groups bar one family. We explain the obstacles in that exceptional case. For [Formula: see text], we classify all finite irreducible subgroups of [Formula: see text]. Our classifications are available publicly in Magma.

COMPUTATIONAL INVESTIGATION OF A4-GRAPHS FOR CERTAIN LEECH LATTICE GROUPS

In the case of a finite simple group , and -conjugacy class of element of order 3, The A4-graph is define as simple graph denoted by A4 has as vertex set and are adjacent if and only if x≠y and xy-1 = yx-1.We aim to investigate computationally the structure of theA4 when Leech Lattice groups.

Investigation of Commuting Graphs for Elements of Order 3 in Certain Leech Lattice Groups

Assume that G is a finite group and X is a subset of G. The commuting graph is denoted by С(G,X) and has a set of vertices X with two distinct vertices x, y Î X, being connected together on the condition of xy = yx. In this paper, we investigate the structure of Ϲ(G,X) when G is a particular type of Leech lattice groups, namely Higman–Sims group HS and Janko group J2, along with X as a G-conjugacy class of elements of order 3. We will pay particular attention to analyze the discs’ structure and determinate the diameters, girths, and clique number for these graphs.

The Lengths of Certain Real Conjugacy Classes and the Related Prime Graph

Let G be a finite group. In this paper, we study how certain arithmetical conditions on the conjugacy class lengths of real elements of G influence the structure of G. In particular, a new type of prime graph is introduced and studied. We obtain a series of theorems which generalize some existed results.

Groups whose vanishing class sizes are prime powers

For a finite group 𝐺, an element is called a vanishing element of 𝐺 if it is a zero of an irreducible character of 𝐺; otherwise, it is called a non-vanishing element. Moreover, the conjugacy class of an element is called a vanishing class if that element is a vanishing element. In this paper, we describe finite groups whose vanishing class sizes are all prime powers, and on the other hand we show that non-vanishing elements of such a group lie in the Fitting subgroup which is a proof of a conjecture mentioned in [I. M. Isaacs, G. Navarro and T. R. Wolf, Finite group elements where no irreducible character vanishes, J. Algebra 222 (1999), 2, 413–423] under this special restriction on vanishing class sizes.