The Normalizer Of Certain Modular Subgroups

1956 ◽  
Vol 8 ◽  
pp. 29-31 ◽  
Author(s):  
Morris Newman
Keyword(s):  
Positive Integer ◽  
Multiplicative Group ◽  
Modular Groups ◽  
Image Position

Introduction. Let G denote the multiplicative group of matriceswhere a, b, c, d are integers and ad — bc = 1. G is one of the well-known modular groups. Let G0(n) denote the subgroup of G characterized by c ≡ 0 (mod n), where n is a positive integer. In this note we determine the normalizer of G0(n) in G, denoted by .

The Number of Non-cyclic Sylow Subgroups of the Multiplicative Group Modulo n

2020 ◽  
pp. 1-12
Author(s):  
Paul Pollack
Keyword(s):  
Positive Integer ◽  
Multiplicative Group ◽  
Asymptotic Density ◽  
Normal Order ◽  
Sylow Subgroups ◽  
Group Of Units ◽  
Euler’S Function ◽  
Image Position ◽  
Group Modulo

Abstract For each positive integer n, let $U(\mathbf {Z}/n\mathbf {Z})$ denote the group of units modulo n, which has order $\phi (n)$ (Euler’s function) and exponent $\lambda (n)$ (Carmichael’s function). The ratio $\phi (n)/\lambda (n)$ is always an integer, and a prime p divides this ratio precisely when the (unique) Sylow p-subgroup of $U(\mathbf {Z}/n\mathbf {Z})$ is noncyclic. Write W(n) for the number of such primes p. Banks, Luca, and Shparlinski showed that for certain constants $C_1, C_2>0$ , $$ \begin{align*} C_1 \frac{\log\log{n}}{(\log\log\log{n})^2} \le W(n) \le C_2 \log\log{n} \end{align*} $$ for all n from a sequence of asymptotic density 1. We sharpen their result by showing that W(n) has normal order $\log \log {n}/\log \log \log {n}$ .

scholarly journals The Divisibility of Divisor Functions

1961 ◽  
Vol 5 (1) ◽  
pp. 35-40 ◽  
Author(s):  
R. A. Rankin
Keyword(s):  
Positive Integer ◽  
Divisor Functions ◽  
Image Position ◽  
Positive Integers

For any positive integers n and v letwhere d runs through all the positive divisors of n. For each positive integer k and real x > 1, denote by N(v, k; x) the number of positive integers n ≦ x for which σv(n) is not divisible by k. Then Watson [6] has shown that, when v is odd,as x → ∞; it is assumed here and throughout that v and k are fixed and independent of x. It follows, in particular, that σ (n) is almost always divisible by k. A brief account of the ideas used by Watson will be found in § 10.6 of Hardy's book on Ramanujan [2].

The Distribution Of Totatives

1955 ◽  
Vol 7 ◽  
pp. 347-357 ◽  
Author(s):  
D. H. Lehmer
Keyword(s):  
Positive Integer ◽  
Prime Factors ◽  
Image Position

This paper is concerned with the numbers which are relatively prime to a given positive integerwhere the p's are the distinct prime factors of n. Since these numbers recur periodically with period n, it suffices to study the ϕ(n) numbers ≤n and relatively prime to n.

scholarly journals An algebraically closed field

1968 ◽  
Vol 9 (2) ◽  
pp. 146-151 ◽  
Author(s):  
F. J. Rayner
Keyword(s):  
Positive Integer ◽  
Power Series ◽  
Formal Power Series ◽  
Characteristic Zero ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Zero Characteristic ◽  
Image Position ◽  
Formal Power ◽  
Puiseux Expansions

Letkbe any algebraically closed field, and denote byk((t)) the field of formal power series in one indeterminatetoverk. Letso thatKis the field of Puiseux expansions with coefficients ink(each element ofKis a formal power series intl/rfor some positive integerr). It is well-known thatKis algebraically closed if and only ifkis of characteristic zero [1, p. 61]. For examples relating to ramified extensions of fields with valuation [9, §6] it is useful to have a field analogous toKwhich is algebraically closed whenkhas non-zero characteristicp. In this paper, I prove that the setLof all formal power series of the form Σaitei(where (ei) is well-ordered,ei=mi|nprt,n∈ Ζ,mi∈ Ζ,ai∈k,ri∈ Ν) forms an algebraically closed field.

scholarly journals An Integral involving a Modified Bessel Function of the Second Kind and an E-function

1953 ◽  
Vol 1 (3) ◽  
pp. 119-120 ◽  
Author(s):  
Fouad M. Ragab
Keyword(s):  
Positive Integer ◽  
Bessel Function ◽  
Modified Bessel Function ◽  
Image Position

§ 1. Introductory. The formula to be established iswhere m is a positive integer,and the constants are such that the integral converges.

scholarly journals Integrals involving products of modified Bessel functions of the second kind

1963 ◽  
Vol 6 (2) ◽  
pp. 70-74 ◽  
Author(s):  
F. M. Ragab
Keyword(s):  
Positive Integer ◽  
Bessel Functions ◽  
Modified Bessel Functions ◽  
Image Position

It is proposed to establish the two following integrals.where n is a positive integer, x is real and positive, μi and ν are complex, and Δ (n; a) represents the set of parameterswhere n is a positive integer and x is real and positive.

Determining a Set from the Cardinalities of its Intersections with Other Sets

1964 ◽  
Vol 16 ◽  
pp. 94-97 ◽  
Author(s):  
David G. Cantor
Keyword(s):  
Positive Integer ◽  
Image Position

Let n be a positive integer and put N = {1, 2, . . . , n}. A collection {S1, S2, . . . , St} of subsets of N is called determining if, for any T ⊂ N, the cardinalities of the t intersections T ∩ Sj determine T uniquely. Let €1, €2, . . . , €n be n variables with range {0, 1}. It is clear that a determining collection {Sj) has the property that the sums

Representations by Hermitian Forms in a Finite Field of Characteristic Two

1977 ◽  
Vol 29 (1) ◽  
pp. 169-179 ◽  
Author(s):  
John D. Fulton
Keyword(s):  
Positive Integer ◽  
Finite Field ◽  
Multiplicative Group ◽  
Group Homomorphism ◽  
Hermitian Forms ◽  
Characteristic Two ◽  
Field Automorphism

Throughout this paper, we let q = 2W,﹜ w a positive integer, and for u = 1 or 2, we let GF(qu) denote the finite field of cardinality qu. Let - denote the involutory field automorphism of GF(q2) with GF(q) as fixed subfield, where ā = aQ for all a in GF﹛q2). Moreover, let | | denote the norm (multiplicative group homomorphism) mapping of GF(q2) onto GF(q), where |a| — a • ā = aQ+1.

Note Upon The Generalized Cayleyan Operator

1949 ◽  
Vol 1 (1) ◽  
pp. 48-56 ◽  
Author(s):  
H. W. Turnbull
Keyword(s):  
Positive Integer ◽  
Original Proof ◽  
Image Position

The following note which deals with the effect of a certain determinantal operator when it acts upon a product of determinants was suggested by the original proof which Dr. Alfred Young gave of the propertysubsisting between the positive P and the negative N substitutional operators, θ being a positive integer. This result which establishes the idempotency of the expression θ−1NP within an appropriate algebra is fundamental in the Quantitative Substitutional Analysis that Young developed.

On the Integral Part of a Linear form with Prime Variables

1966 ◽  
Vol 18 ◽  
pp. 621-628 ◽  
Author(s):  
I. Danicic
Keyword(s):  
Positive Integer ◽  
Linear Form ◽  
Image Position

The object of this paper is to prove the following:Theorem. Suppose that λ, μ are real non-zero numbers, not both negative, λ is irrational, and k is a positive integer. Then there exist infinitely many primes p and pairs of primes p1, p2 such thatIn particular [λp1 + μp2] represents infinitely many primes.Here [x] denotes the greatest integer not exceeding x.

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