We study the computational problem of checking whether a logical sentence is true in a finite abelian group. We prove that model checking first-order sentences on finite abelian groups is fixed-parameter tractable, when parameterized by the size of the sentence. We also prove that model checking monadic second-order sentences on finite abelian groups finitely presented by integer matrices is no...

These are some notes I have written up for myself. I have compiled what I feel to be some of the most important facts concerning characters on finite abelian groups, especially from a number theoretic point of view. I decided to distribute these because I couldn’t find a single reference that covered everything that I thought was relevant to the material I will discuss on Wednesday. Of course, ...

Let G be a finite abelian group of order n. For a complex valued function f on G let f̂ denote the Fourier transform of f . The classical uncertainty inequality asserts that if f 6= 0 then |supp(f)| · |supp(f̂)| ≥ |G| . (1) Answering a question of Terence Tao, the following improvement of (1) is shown: Theorem: Let d1 < d2 be two consecutive divisors of n. If d1 ≤ k = |supp(f)| ≤ d2 then |supp(f̂)...

A regular Cayley map for a finite group A is an orientable map whose orientation-preserving automorphism group G acts regularly on the directed edge set and has a subgroup isomorphic to A that acts regularly on the vertex set. This paper considers the problem of determining which abelian groups have regular Cayley maps. The analysis is purely algebraic, involving the structure of the canonical ...

This article continues a prior investigation of the authors with the goal of extending characterization results of convolutional tight frames from the context of cyclic groups to general finite abelian groups. The collections studied are formed by translating a number of generators by elements of a fixed subgroup and it is shown, under certain norm conditions, that tight frames with this struct...

A set of quasi-uniform random variables X1, . . . , Xn may be generated from a finite group G and n of its subgroups, with the corresponding entropic vector depending on the subgroup structure of G. It is known that the set of entropic vectors obtained by considering arbitrary finite groups is much richer than the one provided just by abelian groups. In this paper, we start to investigate in mo...

Example 1.2. The trivial character of G is the homomorphism 1G defined by 1G(g) = 1 for all g ∈ G. Example 1.3. Let G be cyclic of order 4 with generator γ. Since γ4 = 1, a character χ of G has χ(γ)4 = 1, so χ takes only four possible values at γ, namely 1, −1, i, or −i. Once χ(γ) is known, the value of χ elsewhere is determined by multiplicativity: χ(γj) = χ(γ)j . So we get four characters, wh...

here we consider all finite non-abelian 2-generator $p$-groups ($p$ an odd prime) of nilpotency class two and study the probability of having $n^{th}$-roots of them. also we find integers $n$ for which, these groups are $n$-central.

‎let us consider the set of non-abelian finite simple groups which‎ ‎admit non-trivial irreducible projective representations of degree $le 7$ over‎ ‎an algebraically closed field $f$ of characteristic $pgeq 0$‎. ‎we survey some recent results which‎ ‎lead to the complete list of the groups in this set which are‎ ‎$(2‎, ‎3‎, ‎7)$-generated and of those which are $(2,3)$-generated‎.

‎let us consider the set of non-abelian finite simple groups which‎ ‎admit non-trivial irreducible projective representations of degree $le 7$ over‎ ‎an algebraically closed field $mathbb{f}$ of characteristic $pgeq 0$‎. ‎we survey some recent results which‎ ‎lead to the complete list of the groups in this set which are‎ ‎$(2,3,7)$-generated and of those which are $(2,3)$-generated‎.