Inverse ambiguous functions on some finite non-abelian groups

In Schmitz (Aequ Math 91:373–389, 2017), the first author defines an “inverse ambiguous function” on a group G to be a bijective function \(f:G \rightarrow G\) satisfying the functional equation \(f^{-1}(x) = (f(x))^{-1}\) for all \(x \in G\). Using a simple criterion involving the number of elements in G not equal to their own inverse, the classification of finite abelian groups admitting inverse ambiguous functions is achieved. In this paper we aim to extend the results from (2017) to determine the existence of inverse ambiguous functions on members of certain families of non-abelian groups, namely the symmetric groups \(S_n\), the alternating groups \(A_n\), and the general linear groups GL(2, q) over a finite field \(\mathbb {F}_q\).     

A note on Toeplitz operators on discrete groups

We study Toeplitz algebras associated to partially-ordered and quasi-partially ordered discrete groups.

     

A note on TI-subgroups of finite groups

A subgroup H of a finite group G is called a TI-subgroup if H ∩ H ^x  = 1 or H for any x ∈ G. In this short note, the finite groups all of whose nonabelian subgroups are TI-subgroups are classified.     

A note on differential operators of infinite order

A sufficient condition for entire functions f and g to be such that the series ∑_m=0^∞f^(m)(0)g^(m)/m! represents an entire function is established; and in that case, the growth of the resulting function is described.     

Almost‐periodic solutions to quasilinear evolution equations with a nonlocal nonlinearity

We prove an existence and uniqueness result for almost‐periodic solutions to the quasilinear evolution equations (1) and (5). Copyright © 2006 John Wiley & Sons, Ltd.     

The connes spectrums of certain finite non-abelian groups

In this article, we investigate when every simple module has a projective (pre)envelope. It is proven that (1) every simple right R-module has a projective preenvelope if and only if the left annihilator of every maximal right ideal of R is finitely generated; (2) every simple right R-module has an epic projective envelope if and only if R is a right PS ring; (3) Every simple right R-module has a monic projective preenvelope if and only if R is a right Kasch ring and the left annihilator of every maximal right ideal of R is finitely generated.     

A note on skew differential operators on commutative rings

Let A be a commutative integral domain that is a finitely generated algebra over a field k of characteristic 0 and let ø be a k-algebra automorphism of A of finite order m. In this note we study the ring D(A;ø of differential operators introduced by A.D. Bell. We prove that if A is a free module over the fixed sub-ring A ^ø, with a basis containing 1, then D(A;ø) is isomorphic to the matrix ring M_m(D(A ^ø). It follows from Grothendieck's Generic Flatness Theorem that for an arbitrary A there is an element c?Asuch that D(A[c^-1];ø)?M _m(D(A[c^-1]^ø)). As an application, we consider the structure of D(A;ø)when A is a polynomial or Laurent polynomial ring over k and ø is a diagonalizable linear automorphism.     

A note on groups whose non-normal subgroups are either abelian or minimal non-abelian

A group G is called parahamiltonian if each non-normal subgroup of G is either abelian or minimal non-abelian. Thus all biminimal non-abelian groups are parahamiltonian, and the class of parahamiltonian groups contains the important class of metahamiltonain groups, introduced by Romalis and Sesekin about 50 years ago. The aim of this paper is to describe the structure of locally graded parahamiltonian groups.

     

Some results on minimal sumset sizes in finite non-abelian groups

Let G be a group. We study the minimal sumset (or product set) size μG(r,s)=min{|A⋅B|}, where A,B range over all subsets of G with cardinality r,s respectively. The function μG has recently been fully determined in [S. Eliahou, M. Kervaire, A. Plagne, Optimally small sumsets in finite abelian groups, J. Number Theory 101 (2003) 338-348; S. Eliahou, M. Kervaire, Minimal sumsets in infinite abelian groups, J. Algebra 287 (2005) 449-457] for G abelian. Here we focus on the largely open case where G is finite non-abelian. We obtain results on μG(r,s) in certain ranges for r and s, for instance when r?3 or when r+s?|G|−1, and under some more technical conditions. (See Theorem 4.4.) We also compute μG for a few non-abelian groups of small order. These results extend the Cauchy-Davenport theorem, which determines μG(r,s) for G a cyclic group of prime order.     

A note on 2-nilpotence of finite groups

Ballester-Bolinches and Guo showed that a finite group G is 2-nilpotent if G satisfies: (1) a Sylow 2-subgroup P of G is quaternion-free and (2) Ω₁(P ∩  G′)?≤?Z(P) and N _G (P) is 2-nilpotent. In this paper, it is obtained that G is a non-2-nilpotent group of order 16q for an odd prime q satisfying (1) a Sylow 2-subgroup P of G is not quaternion-free and (2) Ω₁(P?∩?G′)?≤?Z(P) and N _G (P) is 2-nilpotent if and only if q?=?3 and G???GL ₂(3).     

A note on conjugacy classes of finite groups

Let G be a finite group and let x ^G 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.     

A note on weakly-supplemented subgroups of finite groups

A subgroup H of a finite group G is weakly-supplemented in G if there exists a proper subgroup K of G such that G = HK. In the paper, we extend one main result of Kong and Liu (2014).