In this paper,we give a further study of strongly nil Gf-clean rings.Some new characterizations and extension properties of strongly nil Gf-clean rings are provided.Furthermore,motivated by the notion of strong Drazin inverses,we introduce a new type of generalized inverse which is closely related to strongly nil Gf-clean elements,and we present basic properties and structures of this type of inverse.
Let G be a group. A weak Cayley table isomorphism is a bijection phi: G -> G such that (i) phi(g(1))p(g(2)) is conjugate to phi(g(1)g(2)) for g(1), g(2) is an element of G and (ii) phi sends conjugacy classes to conjugacy classes. The set of all such bijections forms a group W(G). We study W(G) for 56 of the 219 three-dimensional crystallographic groups, as well as some other groups. These 56 groups are related to our previous work on wallpaper groups. In this paper we determine for which of the 56 groups W(G) has elements that are not automorphisms nor anti-automorphisms.
Let G be a graph. We say that a hypergraph H is Berge-G if there exists a bijection phi from E(G) to E(H) such that e subset of phi(e) for all e is an element of E(G). For any r-uniform hypergraph H and a real number p >= 1, the p-spectral radius lambda((p))(H) of H is defined as max(x is an element of R)n,parallel to x parallel to(p)=1 r Sigma({i1,...,ir}is an element of E(H)) x(i1) & centerdot; & centerdot; & centerdot; x(ir). We study the p-spectral radius of Berge-G hypergraphs and determine the 3-uniform hypergraphs with maximum p-spectral radius for p >= 1 among Berge-G hypergraphs when G is a tree that looks like Y with n vertices.
A graph is edge-transitive if its automorphism group acts transitively on the set of edges of the graph.In this paper,we classify edge-transitive 8-valent graphs of order 9p for each prime p.
We establish that every 2-local inner derivation on a symmetrizable Kac-Moody Lie algebra g(A) over the field C is a derivation. In addition, we demonstrate that if dim(g(A)/g(A)') >= 2, where g(A)' = [g(A), g(A)] is the derived subalgebra of g(A), then g(A) admits a 2-local derivation which is not a derivation.
We introduce the notions of a Kantor superpair and a Jordan-Kantor super-pair, by using which we construct 5-graded Lie superalgebras. In addition, the relationship between a Jordan-Kantor superpair (J, M) and the corresponding 5-graded Lie superalgebra (J pound, M, D) is described. We also characterize the universal central extension of the 5-graded Lie superalgebra (J pound, M) with D being the inner structure superalgebra of (J, M). Finally, we obtain a realization of a class of 5-graded Lie superalgebras, termed 5-graded Lie superalgebras of type B(0, 1), with the aid of Jordan-Kantor superpairs.
Let G be a finite group and A (sic) G. Then A is called n-decomposable in G if it is a union of n distinct G-conjugacy classes, and denote n by ncc(A). A group G is called X-decomposable with X = {ncc(A) | A (sic) G}. This paper proves that there are only five (isomorphism classes) {1, 2, 5}-decomposable finite groups which are non-perfect.
For a nontrivial p-group P, a maximal chain of P is a chain of subgroups 1=P-0 & lnE;P-1 & lnE;& ctdot;& lnE;P-n=P with |P-i:Pi-1|=p for i=1,2,& mldr;,n, where n is an integer. We determine the structure of finite groups having SS-quasinormality of a maximal chain of Sylow subgroups. We obtain new characterizations of finite p-nilpotent groups and supersolvable groups.
Recently, Est & eacute;lyi et al. investigated a representation Theta(T) of the automorphisms of a connected graph X by beta & times;beta unimodular matrices over {0,+/- 1}, where beta is the Betti number of X, and classified the graphs for which the representation is unfaithful, with two problems left open: (1) What is the smallest dimension d such that AutX is faithfully represented by d & times;d unimodular matrices? (2) Given a finite group G, find the smallest graph X such that G <= AutX is faithfully represented by unimodular matrices. In this article, we prove that AutX has a faithful representation by unimodular matrices of dimension d
The comaximal graph Gamma(R) of a commutative ring R is a simple graph with vertex set R and two distinct vertices a and b of Gamma(R) adjacent if and only if aR +bR = R, where aR is the ideal generated by a in R. In this article, the independent domination polynomial Di(Gamma(Z(n)), x) of Gamma(Z(n)) is discussed, along with its unimo dal and log-concave properties for certain values of n. Some auxiliary results related to Di(Gamma(Z(n)), x) are presented in terms of their zeros. In addition, we determine the independence polynomial I(Gamma(Z(n)), x) of Gamma(Z(n)) for special values of n and provide a general result associated with it. The bounds for the zeros of the polynomial I(Gamma(Z(n)), x) are established, and its log-concave and unimo dal properties are examined.
Consider the type A(2) root system with simple roots { e(1) - e(2) , e(2) - e(3) } , and let be the span of e(1) - e(2) and e(2) - e(3) over R . Let Phi be any irreducible reduced root system but not of type C , with rank >= 3 . Namely, Phi is of type A(n) ( n >= 3 ), B-n ( n >= 3 ), D-n ( n >= 3 ), epsilon(6) , epsilon(7) , epsilon(8) or F-4 . We show that the set of nonzero orthogonal projection vectors of Phi to the plane forms the inverse root system of G2 .
In this paper, we determine the unique graph whose least eigenvalue attains the minimum among all the connected simple graphs with given clique number.
Let W be a class of non-finitely graded Lie algebras related to generalized Witt algebras W (l(1) , l(2) , l(3) ; Gamma ) . We give a full description of the Poisson structures on W. It is shown that there is no non-trivial and associative Poisson algebraic structures on W.
In this paper, we describe explicitly the structure of the derivation algebra and automorphism group of the symplectic oscillator Lie algebra g(n)=sp(2n)& ltimes;h(n) (n >= 3), where sp(2n) is the symplectic Lie algebra and h(n) is the (2n+ 1)-dimensional Heisenberg algebra.
This paper investigates the transposed Poisson structures on the Schr & ouml;dinger algebra Sn associated with ( n+1 ) -dimensional space-time of the Schr & ouml;dinger Lie group. We prove that for n not equal 2 , the algebra Sn admits no nontrivial (1)/(2) -derivations and, consequently, no nontrivial transposed Poisson structures. In contrast, for the case n=2 , we explicitly determine all 12 -derivations and the corresponding transposed Poisson structures on S-2 . Additionally, we demonstrate that S2 admits a nontrivial Hom-Lie structure.
An element x of a finite group G is said to be real in G if xy = x -1 for some y in G, and quasi-central if < x > < y > = < y > < x > for each y in G. When G is a 2-group, Omega 1 (G) <= Z (G) and G = Omega R-2 (G) , we prove that if every real element in G of order 4 is quasi-central in G, then every element c in G of order 4 is real and < c > is normal in G. Further, we show that a group G with a Sylow p-subgroup P is p-nilpotent if and only if N-G (P) is p-nilpotent and, for all x in G \ N-G (P) , one of the following holds: (a) every element in P boolean AND P-x boolean AND G (N)(p) of order p is quasi-central in P, and if p=2 , every real element in P boolean AND P-x boolean AND G (N)(p) of order 4 is quasi-central in P; (b) every element in P boolean AND P-x boolean AND G (N)(p) of order p is quasi-central in P, and if p=2 , [ Omega R-2 (P boolean AND P-x boolean AND G(p)(N) ) , P ] subset of Omega(1) ( P boolean AND G (N)(p) ) boolean OR Z ( P boolean AND G (N)(p) ) ; (c) | Omega(R) ( P boolean AND P-x boolean AND G (N)(p)) | <= p (p-1) .
The classification of irreducible conformal S ( p ) -modules of finite rank was given by H.B. Chen for p not equal 0 . In this paper, we use a different way to obtain the classification and improve the results, which prove that such modules must be of rank 1.
For smooth projective varieties X and Y over a perfect field k of positive characteristic p so that X and Y are derived equivalent, and admitting a closed embedding into Pe, we assume that p >= 3e +1. We prove that if X is ordinary (resp., Hodge-Witt), then Y is ordinary (resp., Hodge-Witt).
In this paper, we show that all Coleman automorphisms of wreath products of some finite groups with trivial centers by any finite group are inner. In particular, the normalizer property holds for these groups.
A signed graph is determined by its adjacency spectrum (resp., Laplacian spectrum) if there is no other non-switching isomorphic signed graph having the same adjacency spectrum (resp., Laplacian spectrum). In particular, a starlike tree can also be interpreted as a signed graph. Oboudi [On the eigenvalues and spectral radius of starlike trees, Aequationes Math. 92 (2018) 683-694] characterized all starlike trees whose adjacency eigenvalues are all in the interval (-2,2), which are S(1,2,2), S(1,2,3), S(1,2,4) and S(1,1,n-3) for n >= 4. In this paper, our focus is the problem of spectral determination of them. We prove that S(1,2,2), S(1,2,3), S(1,2,4) and S(1,1,n-3) for n not equal 8,10,11,13,16 are determined by their adjacency spectra, and characterize all signed graphs which are non-switching isomorphic and adjacency cospectral with S(1,1,n-3) for other cases. Further, we show that S(1,2,2), S(1,2,3), S(1,2,4) and S(1,1,n-3) for n not equal 4 are determined by their Laplacian spectra, and we characterize all signed graphs which are non-switching isomorphic and Laplacian cospectral to S(1,1,1).
