Arithmetical Conditions on Element Orders and Group Structure

Arithmetical Functions Iii: Orders of Magnitude

It may at first be surprising that this is a reasonable – and, in fact, vital – question to ask even for the “elementary” functions f for which we have found exact formulas, e.g. d(n), σ(n), φ(n), μ(n) (and also r2(n), which we have not yet taken the time to write down a formula for but could have based upon our study of the Gaussian integers). What we are running up against is nothing less tha...

Element orders and Sylow structure of finite groups

Group Theory: Graphs Related to Element Orders of Finite Groups

There are several graphs associated with finite groups which are currently of interest in the field. This subject of this project are graphs related to the element orders of a finite group. The best known such graph is the prime graph Γ(G) of the group G and is defined as follows. The vertices are the prime numbers dividing the order (=number of elements) of the group, and two such primes are l...

On the Structure of Periodic Arithmetical Maps

This is an introduction to the algebraic theory of periodic arithmetical maps. The topic is connected with number theory, combinatorics, algebra and analysis. 1. Various Backgrounds I. Combinatorial Background. For n ∈ Z = {1, 2, 3, · · · } and a ∈ R(n) = {0, 1, · · · , n− 1}, let a(n) = a+ nZ = {x ∈ Z : x ≡ a (mod n)} = {· · · , a− n, a, a+ n, · · · } and call it a residue class with modulus n...

Extremal orders of compositions of certain arithmetical functions

We study the exact extremal orders of compositions f(g(n)) of certain arithmetical functions, including the functions σ(n), φ(n), σ∗(n) and φ∗(n), representing the sum of divisors of n, Euler’s function and their unitary analogues, respectively. Our results complete, generalize and refine known results. Mathematics Subject Classification: 11A25, 11N37