Trivial source character tables of , part II Farrell, Niamh; Lassueur, Caroline
Proceedings of the Edinburgh Mathematical Society,
08/2023, Letnik:
66, Številka:
3
Journal Article
Recenzirano
Abstract We compute the trivial source character tables (also called species tables of the trivial source ring) of the infinite family of finite groups $\operatorname{SL}_{2}(q)$ for q even over a ...large enough field of odd characteristics. This article is a continuation of our article Trivial Source Character Tables of $\operatorname{SL}_{2}(q)$ , where we considered, in particular, the case in which q is odd in non-defining characteristic.
Let
$f(x)\in \mathbb {Z}x$
be a nonconstant polynomial. Let
$n\ge 1, k\ge 2$
and c be integers. An integer a is called an f-exunit in the ring
$\mathbb {Z}_n$
of residue classes modulo n if
$\gcd ...(f(a),n)=1$
. We use the principle of cross-classification to derive an explicit formula for the number
${\mathcal N}_{k,f,c}(n)$
of solutions
$(x_1,\ldots ,x_k)$
of the congruence
$x_1+\cdots +x_k\equiv c\pmod n$
with all
$x_i$
being f-exunits in the ring
$\mathbb {Z}_n$
. This extends a recent result of Anand et al. ‘On a question of f-exunits in
$\mathbb {Z}/{n\mathbb {Z}}$
’, Arch. Math. (Basel) 116 (2021), 403–409. We derive a more explicit formula for
${\mathcal N}_{k,f,c}(n)$
when
$f(x)$
is linear or quadratic.
Abstract
Let
τ
(
n
) is the number of all divisors of n and
σ
(
n
) is the number of summation of all divisors n, Cavior, presented the number of all subgroups of the dihedral group is equal by
τ
(
n
...) +
σ
(
n
). We in this paper determines a formula for the number of subgroups, normal and cyclic subgroups of the group G =
D
2
n
×
C
p
= 〈
a
,
b
,
c
|
a
n
=
b
2
=
c
p
,
b
a
b
=
a
−
1,
a
,
c
=
b
,
c
= 1〉, where
p
is an odd prime number.
In this paper we find the Circular Divison of The ℱactor Group cℱ(2 2κ×D4, ℤ)/\(\bar{\rho }\)(2 2κ ×D4) when κ is a prime number, where 2 2κ is denoted to Quaternion group of order 4k, such that for ...each positive integer n, there are two generators X and Y for 2 2κ satisfies Q2k={ X i Y j, 0≤ i ≤ 2κ − 1, j=0, 1} which has the following properties{ X 2k=Y 4=I, Y X k Y -1=X -k} and D4 is the Dihedral group of order 8 is generate by a rotation e of order 4 and reflection f of order 2 then 8 elements of D4 can be written as: {I*, e, e2, e3, f, fe, fe2, fe3}.
...if it checks out, it will go some way towards taming the randomness of prime numbers, whole numbers that cannot be divided evenly by any number except themselves and 1. Solving either the Riemann ...or Landau-Siegel problems would mean that the distribution of prime numbers does not have huge statistical fluctuations. (Number theorist James Maynard at the University of Oxford, UK, won a Fields Medal in July for improving on Zhang's result, among other achievements.) Basketcase The problem Zhang now says he has solved dates back to the turn of the twentieth century, when mathematicians were exploring the distribution of prime numbers.
By using the generalized bilinear operations with a prime number p=3, a (2+1)-dimensional gBK-like equation is introduced. classes of interaction solutions of the (2+1)-dimensional gBK-like equation ...are generated through symbolic computation with maple. Some images were plotted to illustrate the dynamical movement of the solutions with specific values of the involved parameters.
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