On the 2-rank and 4-rank of the class group of some real pure quartic number fields

Let K = ℚ ( p d 2 4 ) $\begin{array}{} \displaystyle (\sqrt[4]{pd^{2}}) \end{array}$ be a real pure quartic number field and k = ℚ( p $\begin{array}{} \displaystyle \sqrt{p} \end{array}$ ) its real quadratic subfield, where p ≡ 5 (mod 8) is a prime integer and d an odd square-free integer coprime to p. In this work, we calculate r 2(K), the 2-rank of the class group of K, in terms of the number of prime divisors of d that decompose or remain inert in ℚ( p $\begin{array}{} \displaystyle \sqrt{p} \end{array}$ ), then we will deduce forms of d satisfying r 2(K) = 2. In the last case, the 4-rank of the class group of K is given too.

On permutable strongly 2-maximal and strongly 3-maximal subgroups

We describe the structure of finite solvable non-nilpotent groups in which every two strongly n-maximal subgroups are permutable (n = 2; 3). In particular, it is shown for a solvable non-nilpotent group G that any two strongly 2-maximal subgroups are permutable if and only if G is a Schmidt group with Abelian Sylow subgroups. We also prove the equivalence of the structure of non-nilpotent solvable groups with permutable 3-maximal subgroups and with permutable strongly 3-maximal subgroups. The last result allows us to classify all finite solvable groups with permutable strongly 3-maximal subgroups, and we describe 14 classes of groups with this property. The obtained results also prove the nilpotency of a finite solvable group with permutable strongly n -maximal subgroups if the number of prime divisors of the order of this group strictly exceeds n (n=2; 3).

ON THE -RANK OF CLASS GROUPS OF DIRICHLET BIQUADRATIC FIELDS

We show that for $100\%$ of the odd, square free integers $n> 0$ , the $4$ -rank of $\text {Cl}(\mathbb{Q} (i, \sqrt {n}))$ is equal to $\omega _3(n) - 1$ , where $\omega _3$ is the number of prime divisors of n that are $3$ modulo $4$ .

The genus of the subgroup graph of a finite group

For a finite group [Formula: see text] denote by [Formula: see text] the genus of the subgroup graph of [Formula: see text] We prove that [Formula: see text] tends to infinity as either the rank of [Formula: see text] or the number of prime divisors of [Formula: see text] tends to infinity.

A new upper bound for numbers with the Lehmer property and its application to repunit numbers

A composite positive integer [Formula: see text] has the Lehmer property if [Formula: see text] divides [Formula: see text] where [Formula: see text] is an Euler totient function. In this paper, we shall prove that if [Formula: see text] has the Lehmer property, then [Formula: see text], where [Formula: see text] is the number of prime divisors of [Formula: see text]. We apply this bound to repunit numbers and prove that there are at most finitely many numbers with the Lehmer property in the set [Formula: see text] where [Formula: see text] denotes the highest power of 2 that divides [Formula: see text], and [Formula: see text] is a fixed real number.

Asymptotic behavior of functions Ω(k; n) and ω(k; n) related to the number of prime divisors

This article is related to the average estimates of numerical functions Ω(k; n) and ω(k; n) connected with the number of prime divisors of n with limited multiplicity.

On Degrees of Modular Common Divisors and the Big PrimegcdAlgorithm

We consider a few modifications of the Big prime modulargcdalgorithm for polynomials inZ[x]. Our modifications are based on bounds of degrees of modular common divisors of polynomials, on estimates of the number of prime divisors of a resultant, and on finding preliminary bounds on degrees of common divisors using auxiliary primes. These modifications are used to suggest improved algorithms forgcdcalculation and for coprime polynomials detection. To illustrate the ideas we apply the constructed algorithms on certain polynomials, in particular on polynomials from Knuth’s example of intermediate expression swell.