scholarly journals Permutations with Orders Coprime to a Given Integer

10.37236/8678 ◽  
2020 ◽  
Vol 27 (1) ◽  
Author(s):  
John Bamberg ◽  
S. P. Glasby ◽  
Scott Harper ◽  
Cheryl E. Praeger
Keyword(s):  
Positive Integer ◽  
Symmetric Group ◽  
Positive Constant ◽  
Unbounded Function ◽  
Euler’S Totient Function

Let $m$ be a positive integer and let $\rho(m,n)$ be the proportion of permutations of the symmetric group $\mathrm{Sym}(n)$ whose order is coprime to $m$. In 2002, Pouyanne proved that $\rho(n,m)n^{1-\frac{\phi(m)}{m}}\sim \kappa_m$ where $\kappa_m$ is a complicated (unbounded) function of $m$. We show that there exists a positive constant $C(m)$ such that, for all $n \geq m$,\[C(m) \left(\frac{n}{m}\right)^{\frac{\phi(m)}{m}-1} \leq \rho(n,m) \leq \left(\frac{n}{m}\right)^{\frac{\phi(m)}{m}-1}\]where $\phi$ is Euler's totient function.

scholarly journals On the Number of Witnesses in the Miller–Rabin Primality Test

Symmetry ◽  
2020 ◽  
Vol 12 (6) ◽  
pp. 890
Author(s):  
Shamil Talgatovich Ishmukhametov ◽  
Bulat Gazinurovich Mubarakov ◽  
Ramilya Gakilevna Rubtsova
Keyword(s):  
Positive Integer ◽  
Euler’S Totient Function ◽  
Number Of Factors ◽  
Primality Test ◽  
Average Probability ◽  
Test Errors ◽  
The Difference

In this paper, we investigate the popular Miller–Rabin primality test and study its effectiveness. The ability of the test to determine prime integers is based on the difference of the number of primality witnesses for composite and prime integers. Let W ( n ) denote the set of all primality witnesses for odd n. By Rabin’s theorem, if n is prime, then each positive integer a < n is a primality witness for n. For composite n, the power of W ( n ) is less than or equal to φ ( n ) / 4 where φ ( n ) is Euler’s Totient function. We derive new exact formulas for the power of W ( n ) depending on the number of factors of tested integers. In addition, we study the average probability of errors in the Miller–Rabin test and show that it decreases when the length of tested integers increases. This allows us to reduce estimations for the probability of the Miller–Rabin test errors and increase its efficiency.

A Class of Trinomials with Galois Group Sn

2012 ◽  
Vol 19 (spec01) ◽  
pp. 905-911 ◽  
Author(s):  
Anuj Bishnoi ◽  
Sudesh K. Khanduja
Keyword(s):  
Positive Integer ◽  
Symmetric Group ◽  
Galois Group ◽  
Rational Numbers ◽  
Alternating Group ◽  
Polynomial Formula

A well known result of Schur states that if n is a positive integer and a0, a1,…,an are arbitrary integers with a0an coprime to n!, then the polynomial [Formula: see text] is irreducible over the field ℚ of rational numbers. In case each ai = 1, it is known that the Galois group of fn(x) over ℚ contains An, the alternating group on n letters. In this paper, we extend this result to a larger class of polynomials fn(x) which leads to the construction of trinomials of degree n for each n with Galois group Sn, the symmetric group on n letters.

scholarly journals On the monoid of cofinite partial isometries of $\qq{N}^n$ with the usual metric

2019 ◽  
Vol 12 (3) ◽  
pp. 51-68
Author(s):  
Oleg Gutik ◽  
Anatolii Savchuk
Keyword(s):  
Positive Integer ◽  
Symmetric Group ◽  
Natural Partial Order ◽  
Group Congruence ◽  
Partial Isometries ◽  
Group Of Units ◽  
Minimum Group ◽  
Positive Integers ◽  
Quotient Semigroup ◽  
Free Semilattice

In this paper we study the structure of the monoid Iℕn ∞ of  cofinite partial isometries of the n-th power of the set of positive integers ℕ with the usual metric for a positive integer n > 2. We describe the group of units and the subset of idempotents of the semigroup Iℕn ∞, the natural partial order and Green's relations on Iℕn ∞. In particular we show that the quotient semigroup Iℕn ∞/Cmg, where Cmg is the minimum group congruence on Iℕn ∞, is isomorphic to the symmetric group Sn and D = J in Iℕn ∞. Also, we prove that for any integer n ≥2 the semigroup Iℕn ∞  is isomorphic to the semidirect product Sn ×h(P∞(Nn); U) of the free semilattice with the unit (P∞(Nn); U)  by the symmetric group Sn.

scholarly journals Minimal permutation representations of the two double covers of Sn

1996 ◽  
Vol 39 (2) ◽  
pp. 285-289
Author(s):  
John Brinkman
Keyword(s):  
Finite Group ◽  
Positive Integer ◽  
Symmetric Group ◽  
Double Cover ◽  
Permutation Representation ◽  
Double Covers ◽  
A Finite Group

Let G be a finite group and denote by µ(G) (see [2]) the least positive integer m such that G has a faithful permutation representation in the symmetric group of degree m. This note considers the value of µ(G) when G is a double cover of the symmetric group.

Linear Turán Numbers of Linear Cycles and Cycle-Complete Ramsey Numbers

2017 ◽  
Vol 27 (3) ◽  
pp. 358-386 ◽  
Author(s):  
CLAYTON COLLIER-CARTAINO ◽  
NATHAN GRABER ◽  
TAO JIANG
Keyword(s):  
Positive Integer ◽  
Positive Constant ◽  
Uniform Hypergraph ◽  
Ramsey Numbers ◽  
Turán Number ◽  
Formula Group ◽  
Image Position ◽  
Linear Hypergraphs

Anr-uniform hypergraph is called anr-graph. A hypergraph islinearif every two edges intersect in at most one vertex. Given a linearr-graphHand a positive integern, thelinear Turán numberexL(n,H) is the maximum number of edges in a linearr-graphGthat does not containHas a subgraph. For each ℓ ≥ 3, letCrℓdenote ther-uniform linear cycle of length ℓ, which is anr-graph with edgese1, . . .,eℓsuch that, for alli∈ [ℓ−1], |ei∩ei+1|=1, |eℓ∩e1|=1 andei∩ej= ∅ for all other pairs {i,j},i≠j. For allr≥ 3 and ℓ ≥ 3, we show that there exists a positive constantc=cr,ℓ, depending onlyrand ℓ, such that exL(n,Crℓ) ≤cn1+1/⌊ℓ/2⌋. This answers a question of Kostochka, Mubayi and Verstraëte [30]. For even ℓ, our result extends the result of Bondy and Simonovits [7] on the Turán numbers of even cycles to linear hypergraphs.Using our results on linear Turán numbers, we also obtain bounds on the cycle-complete hypergraph Ramsey numbers. We show that there are positive constantsa=am,randb=bm,r, depending only onmandr, such that\begin{equation} R(C^r_{2m}, K^r_t)\leq a \Bigl(\frac{t}{\ln t}\Bigr)^{{m}/{(m-1)}} \quad\text{and}\quad R(C^r_{2m+1}, K^r_t)\leq b t^{{m}/{(m-1)}}. \end{equation}

scholarly journals Product of Conjugacy Classes of the Alternating Group An

2012 ◽  
Vol 9 (3) ◽  
pp. 565-568
Author(s):  
Baghdad Science Journal
Keyword(s):  
Positive Integer ◽  
Conjugacy Class ◽  
Symmetric Group ◽  
Nonempty Subset ◽  
Conjugacy Classes ◽  
Alternating Group ◽  
Class C

For a nonempty subset X of a group G and a positive integer m , the product of X , denoted by Xm ,is the set Xm = That is , Xm is the subset of G formed by considering all possible ordered products of m elements form X. In the symmetric group Sn, the class Cn (n odd positive integer) split into two conjugacy classes in An denoted Cn+ and Cn- . C+ and C- were used for these two parts of Cn. This work we prove that for some odd n ,the class C of 5- cycle in Sn has the property that = An n 7 and C+ has the property that each element of C+ is conjugate to its inverse, the square of each element of it is the element of C-, these results were used to prove that C+ C- = An exceptional of I (I the identity conjugacy class), when n=5+4k , k>=0.

scholarly journals A generalization of Menon’s identity with Dirichlet characters

2018 ◽  
Vol 14 (10) ◽  
pp. 2631-2639 ◽  
Author(s):  
Yan Li ◽  
Xiaoyu Hu ◽  
Daeyeoul Kim
Keyword(s):  
Number Theory ◽  
Positive Integer ◽  
Dirichlet Character ◽  
Greatest Common Divisor ◽  
Divisor Function ◽  
Euler’S Totient Function ◽  
Group Of Units ◽  
Dirichlet Characters ◽  
Indian Math ◽  
Sum Formula

The classical Menon’s identity [P. K. Menon, On the sum [Formula: see text], J. Indian Math. Soc.[Formula: see text]N.S.[Formula: see text] 29 (1965) 155–163] states that [Formula: see text] where for a positive integer [Formula: see text], [Formula: see text] is the group of units of the ring [Formula: see text], [Formula: see text] represents the greatest common divisor, [Formula: see text] is the Euler’s totient function and [Formula: see text] is the divisor function. In this paper, we generalize Menon’s identity with Dirichlet characters in the following way: [Formula: see text] where [Formula: see text] is a non-negative integer and [Formula: see text] is a Dirichlet character modulo [Formula: see text] whose conductor is [Formula: see text]. Our result can be viewed as an extension of Zhao and Cao’s result [Another generalization of Menon’s identity, Int. J. Number Theory 13(9) (2017) 2373–2379] to [Formula: see text]. It can also be viewed as an extension of Sury’s result [Some number-theoretic identities from group actions, Rend. Circ. Mat. Palermo 58 (2009) 99–108] to Dirichlet characters.

Radial Entire Solutions of Even Order Semilinear Elliptic Equations

1988 ◽  
Vol 40 (6) ◽  
pp. 1281-1300 ◽  
Author(s):  
Takaŝi Kusano ◽  
Manabu Naito ◽  
Charles A. Swanson
Keyword(s):  
Partial Differential Equations ◽  
Continuous Function ◽  
Positive Integer ◽  
Positive Constant ◽  
Elliptic Equations ◽  
Elliptic Partial Differential Equations ◽  
Entire Solutions ◽  
Semilinear Elliptic ◽  
Even Order ◽  
Image Position

Semilinear elliptic partial differential equations of the type1will be considered throughout real Euclidean N-space, where m ≧ 2 is a positive integer, Δ denotes the N-dimensional Laplacian, and f is a real-valued continuous function in [0, ∞) × (0, ∞). Detailed hypotheses on the structure of f are listed in Section 3.Our objective is to prove the existence of radially symmetric positive entire solutions u(x) of (1) which are asymptotic to positive constant multiples of |x|2m−2i as |x| → ∞ for every i = 1,…, m, N ≧ 2i + 1.

Infinite families of monogenic trinomials and their Galois groups

2018 ◽  
Vol 29 (05) ◽  
pp. 1850039 ◽  
Author(s):  
Lenny Jones ◽  
Tristan Phillips
Keyword(s):  
Positive Integer ◽  
Symmetric Group ◽  
Galois Groups ◽  
Alternating Group

Let [Formula: see text] with [Formula: see text]. Let [Formula: see text] and [Formula: see text] denote, respectively, the symmetric group and alternating group on [Formula: see text] letters. Let [Formula: see text] be an indeterminate, and define [Formula: see text] where [Formula: see text] are certain prescribed forms in [Formula: see text]. For a certain set of these forms, we show unconditionally that there exist infinitely many primes [Formula: see text] such that [Formula: see text] is irreducible over [Formula: see text], [Formula: see text], and the fields [Formula: see text] are distinct and monogenic, where [Formula: see text]. Using a different set of forms, we establish a similar result for all square-free values of [Formula: see text], with [Formula: see text], and any positive integer value of [Formula: see text] for which [Formula: see text] is square-free. Additionally, in this case, we prove that [Formula: see text]. Finally, we show that these results can be extended under the assumption of the [Formula: see text]-conjecture. Our methods make use of recent results of Helfgott and Pasten.

scholarly journals Limit Probabilities for Random Sparse Bit Strings

10.37236/1308 ◽  
1997 ◽  
Vol 4 (1) ◽  
Author(s):  
Katherine St. John
Keyword(s):  
Positive Integer ◽  
Differentiable Function ◽  
Positive Constant ◽  
Linear Order ◽  
Unary Predicate ◽  
First Order ◽  
Infinitely Differentiable Function

Let $n$ be a positive integer, $c$ a real positive constant, and $p(n) = c/n$. Let $U_{n,p}$ be the random unary predicate under the linear order, and $S_c$ the almost sure theory of $U_{n,{c\over n}}$. We show that for every first-order sentence $\phi$: $$ f_{\phi}(c) = \lim_{n\rightarrow\infty}{\Pr}[U_{n,{c\over n}} { has\ property\ } \phi] $$ is an infinitely differentiable function. Further, let $S = \bigcap_c S_c$ be the set of all sentences that are true in every almost sure theory. Then, for every $c>0$, $S_c = S$.

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