A digraph is associated with a finite group by utilizing the power map f: G → G defined by f(x) = xkfor all x \in G, where k is a fixed natural number. It is denoted by γG(n, k). In this paper, the generalized quaternion and 2-groups are stud- ied. The height structure is discussed for the generalized quaternion. The necessary and sufficient conditions on a power digraph of a 2-group are determined for a 2-group to be a generalized quaternion group. Further, the classification of two generated 2-groups as abelian or non-abelian in terms of semi-regularity of the power digraphs is completed., Uzma Ahmad, Muqadas Moeen., and Obsahuje seznam literatury
Let M be an m-dimensional manifold and A = Dr k/I = R⊕NA a Weil algebra of height r. We prove that any A-covelocity TA x f ∈ TA x M, x ∈ M is determined by its values over arbitrary max{widthA,m} regular and under the first jet projection linearly independent elements of TA x M. Further, we prove the rigidity of the so-called universally reparametrizable Weil algebras. Applying essentially those partial results we give the proof of the general rigidity result TA M ≃ T r M without coordinate computations, which improves and generalizes the partial result obtained in Tomáš (2009) from m > k to all cases of m. We also introduce the space JA(M,N) of A-jets and prove its rigidity in the sense of its coincidence with the classical jet space Jr(M,N)., Jiří Tomáš., and Seznam literatury
Let Ln = K[x1±1,..., xn±1] be a Laurent polynomial algebra over a field K of characteristic zero, Wn:= DerK(Ln) the Lie algebra of K-derivations of the algebra Ln, the so-called Witt Lie algebra, and let Vir be the Virasoro Lie algebra which is a 1-dimensional central extension of the Witt Lie algebra. The Lie algebras Wn and Vir are infinite dimen- sional Lie algebras. We prove that the following isomorphisms of the groups of Lie algebra automorphisms hold: AutLie(Vir) \simeq AutLie(W1) \simeq {±1} \simeq K*, and give a short proof that AutLie(Wn) \simeq AutK-alg(Ln) \simeq GLn(Z) \ltimes K*n., Vladimir V. Bavula., and Obsahuje seznam literatury
Let X be a Stein manifold of complex dimension n\geqslant 2 and \Omega \Subset X be a relatively compact domain with C^{2} smooth boundary in X. Assume that Ω is a weakly q-pseudoconvex domain in X. The purpose of this paper is to establish sufficient conditions for the closed range of \overline \partial on Ω. Moreover, we study the \overline \partial -problem on Ω. Specifically, we use the modified weight function method to study the weighted \overline \partial -problem with exact support in Ω. Our method relies on the L^{2} -estimates by Hörmander (1965) and by Kohn (1973)., Sayed Saber., and Obsahuje seznam literatury
Suppose that A is a real symmetric matrix of order n. Denote by m_{A}(0) the nullity of A. For a nonempty subset α of {1, 2,..., n}, let A(α) be the principal submatrix of A obtained from A by deleting the rows and columns indexed by α. When m_{A(\alpha )}(0) = m_{A}(0)+|α|, we call α a P-set of A. It is known that every P-set of A contains at most \left \lfloor n/2 \right \rfloorelements. The graphs of even order for which one can find a matrix attaining this bound are now completely characterized. However, the odd case turned out to be more difficult to tackle. As a first step to the full characterization of these graphs of odd order, we establish some conditions for such graphs G under which there is a real symmetric matrix A whose graph is G and contains a P-set of size (n − 1)/2., Zhibin Du, Carlos M. da Fonseca., and Obsahuje seznam literatury
For a finite group $G$ denote by $N(G)$ the set of conjugacy class sizes of $G$. In 1980s, J. G. Thompson posed the following conjecture: If $L$ is a finite nonabelian simple group, $G$ is a finite group with trivial center and $N(G) = N(L)$, then $G\cong L$. We prove this conjecture for an infinite class of simple groups. Let $p$ be an odd prime. We show that every finite group $G$ with the property $Z(G)=1$ and $N(G) = N(A_i)$ is necessarily isomorphic to $A_i$, where $i\in\{2p,2p+1\}$., Azam Babai, Ali Mahmoudifar., and Obsahuje bibliografii
We investigate the Zassenhaus conjecture regarding rational conjugacy of torsion units in integral group rings for certain automorphism groups of simple groups. Recently, many new restrictions on partial augmentations for torsion units of integral group rings have improved the effectiveness of the Luther-Passi method for verifying the Zassenhaus conjecture for certain groups. We prove that the Zassenhaus conjecture is true for the automorphism group of the simple group PSL(2, 11). Additionally we prove that the Prime graph question is true for the automorphism group of the simple group PSL(2, 13)., Joe Gildea., and Obsahuje seznam literatury
Let SP be the set of upper strongly porous at 0 subsets of \mathbb{R}^{+} and let Î(SP) be the intersection of maximal ideals I\subseteq SP. Some characteristic properties of sets E \in Î(SP) are obtained. We also find a characteristic property of the intersection of all maximal ideals contained in a given set which is closed under subsets. It is shown that the ideal generated by the so-called completely strongly porous at 0 subsets of \mathbb{R}^{+} is a proper subideal of Î(SP). Earlier, completely strongly porous sets and some of their properties were studied in the paper V.Bilet, O.Dovgoshey (2013/2014)., Viktoriia Bilet, Oleksiy Dovgoshey, Jürgen Prestin., and Obsahuje seznam literatury