Let G be a finite classical group, specifically one of the general linear, unitary, symplectic, or orthogonal groups defined over a finite field of odd characteristic. For any element g ∈ G, we consider the fiber of the squaring map at g, which is the set of all elements h ∈ G satisfying h2 = g. In this work, we determine the cardinality of these fibers for arbitrary elements g ∈ G. Our analysis enables us to compute the total number of real conjugacy classes in G. The central tool for this is a recent resolution of Brauers Problem 14.

Let A be a finite group (or a finite algebra), and ω be a word map (resp. polynomial map) on n many generators. We define the quantity |ω(A)|/|A| as the image ratio ofωonA and denote it by μ(ω,A). In this article, we investigate the set R(ω)={μ(ω,A):Ais a finite group}, and study the same for the case of rings. We demonstrate the existence of word maps whose set of image ratios is dense in [0, 1] for groups (and rings).

Let (Formula presented.) be the split octonion algebra over an algebraically closed field (Formula presented.). For positive integers (Formula presented.), we study surjectivity of the map (Formula presented.) on (Formula presented.). For this, we use the orbit representatives of the (Formula presented.) -action on (Formula presented.) for the tuple (Formula presented.), and characterize the ones which give a surjective map.

For n⩾2, we consider the polynomial maps on Mn(K) given by evaluation of a polynomial f(X1,…,Xm) over the field K. We explore the image of the diagonal map given by in terms of the solution of certain equations over K. We show that when K=R and m=2, it is surjective except when n is odd, δ1δ2>0, and k1,k2 are both even (in that case, the image misses negative scalars), and the map is surjective for m⩾3. We further show that on Mn(H) (even with H coefficients) the diagonal map is surjective for m⩾2, where H is the algebra of Hamilton’s quaternions.

Let (Formula presented.) be the (Formula presented.) matrix algebra over (Formula presented.) and (Formula presented.) be the invertible elements in (Formula presented.). Inspired by Kaplansky–Lv́ov conjecture, we explore the image of polynomials with constants, namely polynomials from the free algebra (Formula presented.). In this article, we compute the images of the polynomial maps given by (a) generalized sum of powers (Formula presented.) and (b) generalized commutator map (Formula presented.), where (Formula presented.), (Formula presented.) are nonzero elements of (Formula presented.) when (Formula presented.) is an algebraically closed field. We show that the images of these maps are vector spaces. For the polynomial in (a), we compute the images by fixing a simultaneous conjugate pair for (Formula presented.), and it turns out that it is surjective in most cases.

For an integer M ≥ 2 and a finite group G, an element α ∈ G is called an M-th power if it satisfies AM = α for some A ∈ G. In this article, we will deal with the case when G is a finite symplectic or orthogonal group over a field of odd order q. We introduce the notion of M*-power SRIM polynomials. This, amalgamated with the concept of M-power polynomial, we provide the complete classification of the conjugacy classes of regular semisimple, semisimple, cyclic and regular elements in G, which are M-th powers, when (M, q) = 1. The approach here is of generating functions, as worked on by Jason Fulman, Peter M. Neumann, and Cheryl Praeger in the memoir “A generating function approach to the enumeration of matrices in classical groups over finite fields”. As a byproduct, we obtain the corresponding probabilities, in terms of generating functions.

Let q be an odd prime, and let Gn(q) denote one of the finite classical groups—namely, the general linear, unitary, symplectic, or orthogonal group—of rank n over the field GF(q) with q elements. Given an integer M≥2, an element g∈Gn(q) is said to be a root of identity if gM=1. In this article, we derive generating functions for the probability that a randomly chosen element of Gn(q) is an M-th root of identity. Our results depend on the specific types of irreducible factors that appear in the factorization of the polynomial xM-1 over GF(q).

For a group G and a character χ of G, let c(χ) denote the set of all irreducible characters of G, occurring in χ. The character covering number of G is defined as the least n such that c(χn) = Irr(G), for all faithful irreducible χ. In this article, we compute the character covering number of PSL2(q) for all q ≥ 8.

Let U(n,Fq2) denote the subgroup of unitary matrices of the general linear group GL(n,Fq2) which fix a Hermitian form and let M≥2 be an integer. In earlier work, elements of the groups GL(n,Fq), Sp(2n,Fq), O±(2n,Fq) and O(2n+1,Fq) with an M-th root have been described. Here we will describe the M-th powers in unitary groups for the regular semisimple, semisimple and cyclic elements, under the assumption that M and q are coprime. We will further describe the generating functions in the corresponding cases.

Let n≥1 be an integer, p,q be distinct odd primes. Let G,N be two groups of order pnq with their Sylow-p-subgroups being cyclic. We enumerate the Hopf-Galois structures on a Galois G-extension, with type N. This also computes the number of skew braces with additive group isomorphic to N and multiplicative group isomorphic to G. Further when q<p, we give a complete classification of the Hopf-Galois structures on Galois-G-extensions.

It was conjectured that the augmentation ideal of a dihedral quandle of even order n > 2 satisfies (Formula presented) for all k ≥ 2. In this article we provide a counterexample against this conjecture.

Let p be a polynomial in non-commutative variables x1,x2,…,xn with constant term zero over an algebraically closed field K. The object of study in this paper is the image of this kind of polynomial over the algebra of upper triangular matrices Tm(K). We introduce a family of polynomials called multi-index p-inductive polynomials for a given polynomial p. Using this family we will show that, if p is a polynomial identity of Tt(K) but not of Tt+1(K), then p(Tm(K))⊆Tm(K)(t−1). Equality is achieved in the case t=1,m−1 and an example has been provided to show that equality does not hold in general. We further prove existence of d such that each element of Tm(K)(t−1) can be written as sum of d many elements of p(Tm(K)). It has also been shown that the image of Tm(K)× under a word map is Zariski dense in Tm(K)×.

A group G is called to be acceptable (due to M. Larsen) if for any finite group H, two element-conjugate homomorphisms are globally conjugate. We prove that any semisimple algebraic group over an algebraically closed field of non-zero characteristics is not acceptable.

Let G and N be finite groups of order 2n where n is odd. We say the pair (G,N) is Hopf-Galois realizable if G is a regular subgroup of Hol(N)=N⋊Aut(N). In this article we give necessary conditions on G (similarly N) when N (similarly G) is a group of the form Zn⋊Z2, for (G,N) to be realizable. Further we show that this condition is also sufficient if radical of n is a Burnside number. This classifies all skew braces which have the additive group (or the multiplicative group) isomorphic to Zn⋊Z2, in this case.

Let K/F be a finite Galois extension of fields with Gal(K/F)=Γ. In an earlier work of Timothy Kohl, the author enumerated dihedral Hopf-Galois structures acting on dihedral extensions. The dihedral group is one particular example of a semidirect product of Zn and Z2. In this article we count the number of Hopf-Galois structures with Galois group Γ of type G, where Γ,G are groups of the form Zn⋊ϕZ2 when n is odd with radical of n being a Burnside number. As an application we also find the corresponding number of skew braces.