On $$ \mathfrak{F}_h $$-normal subgroups of finite groups

Let G be a finite group, and let $ \mathfrak{F} $ \mathfrak{F} be a formation of finite groups. We say that a subgroup H of G is $ \mathfrak{F}_h $ \mathfrak{F}_h -normal in G if there exists a normal subgroup T of G such that HT is a normal Hall subgroup of G and (H ∩ T)H _G /H _G is contained in the $ \mathfrak{F} $ \mathfrak{F} -hypercenter $ Z_\infty ^\mathfrak{F} $ Z_\infty ^\mathfrak{F} (G/H _G ) of G/H _G . In this paper, we obtain some results about the $ \mathfrak{F}_h $ \mathfrak{F}_h -normal subgroups and then use them to study the structure of finite groups.     

On \mathfrak{F}-reachable subgroups of finite groups

The structure of $\mathfrak{F}$ -reachable subgroups in Θ-Frattini extensions is established.     

Finite solvable groups whose $$\mathfrak{F}$$ -hypercenter contains all minimal subgroups

Archiv der Mathematik -     

On codes over $$\mathbb {F}_{q}+v\mathbb {F}_{q}+v^{2}\mathbb {F}_{q}$$

In this paper we investigate linear codes with complementary dual (LCD) codes and formally self-dual codes over the ring \(R=\mathbb {F}_{q}+v\mathbb {F}_{q}+v^{2}\mathbb {F}_{q}\), where \(v^{3}=v\), for q odd. We give conditions on the existence of LCD codes and present construction of formally self-dual codes over R. Further, we give bounds on the minimum distance of LCD codes over \(\mathbb {F}_q\) and extend these to codes over R.     

On $\mathfrak{F}_\tau$-$s$-Supplemented Subgroups of Finite Groups

Let $\mathfrak{F}$ be a non-empty formation of groups, $\tau$ a subgroup functor and $H$ a $p$-subgroup of a finite group $G.$ Let $\overline{G}=G/H_G$ and $\overline{H} =H/H_G.$ We say that $H$ is $\mathfrak{F}_\tau$-$s$-supplemented in $G$ if for some subgroup $\overline{T}$ and some $\tau$-subgroup $\overline{S}$ of $\overline{G}$ contained in $\overline{H},$ $\overline{H}\overline{T}$ is subnormal in $\overline{G}$ and $\overline{H} ∩ \overline{T} ≤ \overline{S}Z_{\mathfrak{F}}(\overline{G}).$ In this paper, we investigate the influence of $\mathfrak{F}_\tau$-$s$-supplemented subgroups on the structure of finite groups. Some new characterizations about solubility of finite groups are obtained.     

Cyclic DNA codes over the ring $$\mathbb {F}_2+u\mathbb {F}_2+v\mathbb {F}_2+uv\mathbb {F}_2+v^2\mathbb {F}_2+uv^2\mathbb {F}_2$$

We study the structure of cyclic DNA codes of odd length over the finite commutative ring \(R=\mathbb {F}_2+u\mathbb {F}_2+v\mathbb {F}_2+uv\mathbb {F}_2 + v^2\mathbb {F}_2+uv^2\mathbb {F}_2,~u^2=0, v^3=v\), which plays an important role in genetics, bioengineering and DNA computing. A direct link between the elements of the ring R and 64 codons used in the amino acids of living organisms is established by introducing a Gray map from R to \(R_1=\mathbb {F}_2+u\mathbb {F}_2 ~(u^2=0)\). The reversible and the reversible-complement codes over R are investigated. We also discuss the binary image of the cyclic DNA codes over R. Among others, some examples of DNA codes obtained via Gray map are provided.     

Quantum algebras with representation ring of $$ \mathfrak{s}\mathfrak{l}_2 $$ type

A reducible representation of the Temperley-Lieb algebra is constructed on a tensor product of n-dimensional spaces. As a centralizer of this action, we obtain a quantum algebra (quasi-triangular Hopf algebra) with the representation ring that is equivalent to the representation ring of the Lie algebra. Bibliography: 23 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 347, 2007, pp. 167–177.     

On some permutation binomials and trinomials over $$\mathbb {F}_{2^n}$$

In this work, we completely characterize (1) permutation binomials of the form \(x^{{{2^n -1}\over {2^t-1}}+1}+ ax \in \mathbb {F}_{2^n}[x], n = 2^st, a \in \mathbb {F}_{2^{2t}}^{*}\), and (2) permutation trinomials of the form \(x^{2^s+1}+x^{2^{s-1}+1}+\alpha x \in \mathbb {F}_{2^t}[x]\), where s, t are positive integers. The first result, which was our primary motivation, is a consequence of the second result. The second result may be of independent interest.     

On Projective Modules in Category \mathcal{O}_{int} of Quantum \mathfrak{sl}_{2}

The restriction of a Verma module of ${\bf U}(\mathfrak{sl}_3)$ to ${\bf U}(\mathfrak{sl}_2)$ is isomorphic to a Verma module tensoring with all the finite dimensional simple modules of ${\bf U}(\mathfrak{sl}_2)$ . The canonical basis of the Verma module is compatible with such a decomposition. An explicit decomposition of the tensor product of the Verma module of highest weight 0 with a finite dimensional simple module into indecomposable projective modules in the category $\mathcal O_{\rm{int}}$ of quantum $\mathfrak{sl}_2$ is given.     

The \mathfrak {sl}_{3}-web algebra

In this paper we use Kuperberg’s $\mathfrak {sl}_3$ -webs and Khovanov’s $\mathfrak {sl}_3$ -foams to define a new algebra $K^S$ , which we call the $\mathfrak {sl}_3$ -web algebra. It is the $\mathfrak {sl}_3$ analogue of Khovanov’s arc algebra. We prove that $K^S$ is a graded symmetric Frobenius algebra. Furthermore, we categorify an instance of $q$ -skew Howe duality, which allows us to prove that $K^S$ is Morita equivalent to a certain cyclotomic KLR-algebra of level 3. This allows us to determine the split Grothendieck group $K^{\oplus }_0(\mathcal {W}^S)_{\mathbb {Q}(q)}$ , to show that its center is isomorphic to the cohomology ring of a certain Spaltenstein variety, and to prove that $K^S$ is a graded cellular algebra.     

The Abelian category of quotients of $$\mathcal{L}\mathfrak{F}$$ -spaces

We construct the category of quotients of -spaces and we show that it is Abelian. This answers a question of L. Waelbroeck from 1990.