Using an adaptation of Qin Jiushao’s method from the 13th century, it is possible to prove that a system of linear modular equations ai1xi + · · · + ainxn = ~bi mod ~ mi, i = 1, . . . , n has integer solutions if mi > 1 are pairwise relatively prime and in each row, at least one matrix element aij is relatively prime to mi. The Chinese remainder theorem is the special case, where A has only one...

A group G is simple if and only if the diagonal subgroup of G ×G is a maximal subgroup. This striking property is very easy to prove and raises the question of determining all the maximal subgroups of G , where G denotes the direct product of n copies of G . The first purpose of this note is to answer completely this question. We show in particular that if G is perfect, then any maximal subgrou...

Let G be a finite group and let k be an algebraically closed field of prime characteristic p. Let 5 b e a block of kG, P a /^-subgroup of G and n a /(-element of G. Then the multiplicity of P as a lower defect group of B for the /^-section of n is defined [1, 2, 4] and is denoted by mB(P). We shall prove the following induction theorem for mB(P) in terms of the Brown complex A of chains of non-...

We state a qualitative form of strong multiplicity one for GL1. We derive refinements of strong multiplicity one for automorphic representations arising from Eisenstein series associated to a Borel subgroup on GL(n), and for the cuspidal representations on GL(n) induced from idele class characters of cyclic extensions of prime degree. These results are in accordance with a conjecture of D. Rama...

We prove that, for a positive integer n and subgroup H of automorphisms of a cyclic group Z of order n, there is up to isomorphism a unique connected circulant digraph based on Z admitting an arc-transitive action of ZzH. We refine the Kovács–Li classification of arc-transitive circulants to determine those digraphs with automorphism group larger than ZzH. As an application we construct, for ea...

We prove the following theorem: Let F be an algebraic closure of a finite field of characteristic p. If ρ is a continuous homomorphism from the absolute Galois group of Q to GLn(F) which is isomorphic to a direct sum of one-dimensional representations, and if p > n+1 and the conductors of the one-dimensional representations are pairwise relatively prime, and if a certain parity condition holds,...

ardy and Littlewood conjectured that every large integer $n$ that is not a square is the sum of a prime and a square. They believed that the number $mathcal{R}(n)$ of such representations for $n = p+m^2$ is asymptotically given by begin{equation*} mathcal{R}(n) sim frac{sqrt{n}}{log n}prod_{p=3}^{infty}left(1-frac{1}{p-1}left(frac{n}{p}right)right), end{equation*} where $p$ is a prime, $m$ is a...

Eric Lander conjectured that if G is an abelian group of order v containing a difference set of order n and p is a prime dividing v and n, then the Sylow p-subgroup of G cannot be cyclic. This paper verifies a version of this conjecture for k < 6500. A special case of this version is the non-existence of Menon-Hadamard-McFarland difference sets in 2-groups. We also give an algorithm that easily...