Let $p$ be an odd prime and $q$ a power of $p$. We examine the deformation theory reducible indecomposable Galois representations $\bar{\rho}:G_{\mathbb{Q}}\rightarrow \text{GSp}_{2n}(\mathbb{F}_q)$ that are unramified outside finite set primes $S$ whose image lies in Borel subgroup. show under some additional hypotheses, such have geometric lifts to Witt vectors $\text{W}(\mathbb{F}_q)$. The m...

Abstract The homotopy types of gauge groups principal $$\mathrm{SO}(4)$$ SO ( 4 ) -bundles over $$S^{4}$$ S are classified p -locally for every prime , and partial results obtained integrally. met...

let $n$ be a submodule of a module $m$ and a minimal primary decomposition of $n$ is known‎. ‎a formula to compute baer's lower nilradical of $n$ is given‎. ‎the relations between classical prime submodules and their nilradicals are investigated‎. ‎some situations in which semiprime submodules can be written as finite intersection of classical prime submodule are stated‎.

Remark 0.1 (Notation). |G| denotes the order of a finite group G. [E : F ] denotes the degree of a field extension E/F. We write H ≤ G to mean that H is a subgroup of G, and N G to mean that N is a normal subgroup of G. If E/F and K/F are two field extensions, then when we say that K/F is contained in E/F , we mean via a homomorphism that fixes F. We assume the following basic facts in this set...

Let G denote the projective special linear group PSL(2, q), for a prime power q. It is shown that a finite 2-subgroup of the group V(ZG) of augmentation 1 units in the integral group ring ZG of G is isomorphic to a subgroup of G. Furthermore, it is shown that a composition factor of a finite subgroup of V(ZG) is isomorphic to a subgroup of G.

Let σ={σi:i∈I} be a partition of the set all prime numbers. A subgroup H finite group G is said to σ-subnormal in if can joined by chain subgroups H=H0⊆H1⊆⋯⊆Hn=G where, for every j=1,⋯,n, Hj−1 normal Hj or Hj/CoreHj(Hj−1) σi-group some i∈I. B soluble normalising Nσ-residual non-σ-subnormal G, where Nσ saturated formation σ-nilpotent groups. We show that normalises does not have section σ-residu...

Let G be a finite group. Yakov Berkovic investigated the following concept: A subgroup H of G is called NR−subgroup with respect to G if A = AG ⋂ H for any subgroup A H.In particulary,called a finite group G NN−group if its any subgroup is either normal subgroup or NR−subgroup of G. In fact, all groups with order p -p3 are NN -group,where p is a prime. In this paper, the nature and structure of...

These notes are based on two lectures which were given by the author at the Workshop on Arithmetic Algebraic Geometry, I.I.T. Guwhati, India, fall 2008. They were intended to give a quick survey on classical complex modular forms and their l-adic Galois representations. Accordingly, the focus is rather on results and examples whereas proofs are mainly sketched or omitted. Nothing in these notes...

Let q=2n, and let E/Fqℓ be a generalized Galbraith–Lin–Scott (GLS) binary curve, with ℓ≥2 (ℓ,n)=1. We show that the GLS endomorphism on induces an efficient Jacobian JacH(Fq) of genus-g hyperelliptic curve H corresponding to image GHS Weil-descent attack applied E/Fqℓ, this yields factor-n speedup when using standard index-calculus procedures for solving Discrete Logarithm Problem (DLP) JacH(Fq...

In this paper, we explicitly describe all the elements of sequence fractional parts {af(n)/n}, n=1,2,3,…, where f(x)∈Z[x] is a nonconstant polynomial with positive leading coefficient and a≥2 an integer. We also show that each value w={af(n)/n}, n≥nf nf least integer such f(n)≥n/2 for every n≥nf, attained by infinitely many terms sequence. These results combined some earlier estimates on gaps b...