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1.
In this paper we study the family of finite groups with the property that every maximal abelian normal subgroup is self-centralizing. It is well known that this family contains all finite supersolvable groups, but it also contains many other groups. In fact, every finite group G is a subgroup of some member \(\Gamma \) of this family, and we show that if G is solvable, then \(\Gamma \) can be chosen so that every abelian normal subgroup of G is contained in some self-centralizing abelian normal subgroup of \(\Gamma \).  相似文献   

2.
In this paper we continue to develop the topological method to get semigroup generators of semi-simple Lie groups. Consider a subset \(\Gamma \subset G\) that contains a semi-simple subgroup \(G_{1}\) of G. If one can show that \( \Gamma \) does not leave invariant a contractible subset on any flag manifold of G, then \(\Gamma \) generates G if \(\mathrm {Ad}\left( \Gamma \right) \) generates a Zariski dense subgroup of the algebraic group \(\mathrm {Ad}\left( G\right) \). The proof is reduced to check that some specific closed orbits of \(G_{1}\) in the flag manifolds of G are not trivial in the sense of algebraic topology. Here, we consider three different cases of semi-simple Lie groups G and subgroups \(G_{1}\subset G\).  相似文献   

3.
A path in an edge-colored graph is called rainbow if any two edges of the path have distinct colors. An edge-colored graph is called rainbow connected if there exists a rainbow path between every two vertices of the graph. For a connected graph G, the minimum number of colors that are needed to make G rainbow connected is called the rainbow connection number of G, denoted by rc(G). In this paper, we investigate the relation between the rainbow connection number and the independence number of a graph. We show that if G is a connected graph without pendant vertices, then \(\mathrm{rc}(G)\le 2\alpha (G)-1\). An example is given showing that the upper bound \(2\alpha (G)-1\) is equal to the diameter of G, and so the upper bound is sharp since the diameter of G is a lower bound of \(\mathrm{rc}(G)\).  相似文献   

4.
Let \(X=G/K\) be a symmetric space of noncompact type and rank \(k\ge 2\). We prove that horospheres in X are Lipschitz \((k-2)\)-connected if their centers are not contained in a proper join factor of the spherical building of X at infinity. As a consequence, the distortion dimension of an irreducible \(\mathbb {Q}\)-rank-1 lattice \(\Gamma \) in a linear, semisimple Lie group G of \(\mathbb R\)-rank k is \(k-1\). That is, given \(m< k-1\), a Lipschitz m-sphere S in (a polyhedral complex quasi-isometric to) \(\Gamma \), and a \((m+1)\)-ball B in X (or G) filling S, there is a \((m+1)\)-ball \(B'\) in \(\Gamma \) filling S such that \({{\mathrm{vol}}}B'\sim {{\mathrm{vol}}}B\). In particular, such arithmetic lattices satisfy Euclidean isoperimetric inequalities up to dimension \(k-1\).  相似文献   

5.
An automorphism \(\alpha \) of a Cayley graph \(\mathrm{Cay}(G,S)\) of a group G with connection set S is color-preserving if \(\alpha (g,gs) = (h,hs)\) or \((h,hs^{-1})\) for every edge \((g,gs)\in E(\mathrm{Cay}(G,S))\). If every color-preserving automorphism of \(\mathrm{Cay}(G,S)\) is also affine, then \(\mathrm{Cay}(G,S)\) is a Cayley color automorphism (CCA) graph. If every Cayley graph \(\mathrm{Cay}(G,S)\) is a CCA graph, then G is a CCA group. Hujdurovi? et al. have shown that every non-CCA group G contains a section isomorphic to the non-abelian group \(F_{21}\) of order 21. We first show that there is a unique non-CCA Cayley graph \(\Gamma \) of \(F_{21}\). We then show that if \(\mathrm{Cay}(G,S)\) is a non-CCA graph of a group G of odd square-free order, then \(G = H\times F_{21}\) for some CCA group H, and \(\mathrm{Cay}(G,S) = \mathrm{Cay}(H,T)\mathbin {\square }\Gamma \).  相似文献   

6.
Let \(\Gamma \) be a distance-regular graph with diameter d and Kneser graph \(K=\Gamma _d\), the distance-d graph of \(\Gamma \). We say that \(\Gamma \) is partially antipodal when K has fewer distinct eigenvalues than \(\Gamma \). In particular, this is the case of antipodal distance-regular graphs (K with only two distinct eigenvalues) and the so-called half-antipodal distance-regular graphs (K with only one negative eigenvalue). We provide a characterization of partially antipodal distance-regular graphs (among regular graphs with \(d+1\) distinct eigenvalues) in terms of the spectrum and the mean number of vertices at maximal distance d from every vertex. This can be seen as a more general version of the so-called spectral excess theorem, which allows us to characterize those distance-regular graphs which are half-antipodal, antipodal, bipartite, or with Kneser graph being strongly regular.  相似文献   

7.
Gagola and Lewis proved that a finite group G is nilpotent if and only if \(\chi (1)^2\) divides |G :  \(\mathrm{Ker}\) \(\chi |\) for all irreducible characters \(\chi \) of G. In this paper, we prove the following generalization that a finite group G is nilpotent if and only if \(\chi (1)^2\) divides |G :  \(\mathrm{Ker}\) \(\chi |\) for all monolithic characters \(\chi \) of G.  相似文献   

8.
Let R be a commutative ring with a nonzero identity element. For a natural number n, we associate a simple graph, denoted by \(\Gamma ^n_R\), with \(R^n\backslash \{0\}\) as the vertex set and two distinct vertices X and Y in \(R^n\) being adjacent if and only if there exists an \(n\times n\) lower triangular matrix A over R whose entries on the main diagonal are nonzero and one of the entries on the main diagonal is regular such that \(X^TAY=0\) or \(Y^TAX=0\), where, for a matrix \(B, B^T\) is the matrix transpose of B. If \(n=1\), then \(\Gamma ^n_R\) is isomorphic to the zero divisor graph \(\Gamma (R)\), and so \(\Gamma ^n_R\) is a generalization of \(\Gamma (R)\) which is called a generalized zero divisor graph of R. In this paper, we study some basic properties of \(\Gamma ^n_ R\). We also determine all isomorphic classes of finite commutative rings whose generalized zero divisor graphs have genus at most three.  相似文献   

9.
In this paper, we first give a classification of the family of 2-geodesic transitive abelian Cayley graphs. Let \(\Gamma \) be such a graph which is not 2-arc transitive. It is shown that one of the following holds: (1) \(\Gamma \cong \mathrm{K}_{m[b]}\) for some \(m\ge 3\) and \(b\ge 2\); (2) \(\Gamma \) is a normal Cayley graph of an elementary abelian group; (3) \(\Gamma \) is a cover of Cayley graph \(\Gamma _K\) of an abelian group T / K, where either \(\Gamma _K\) is complete arc transitive or \(\Gamma _K\) is 2-geodesic transitive of girth 3, and A / K acts primitively on \(V(\Gamma _K)\) of type Affine or Product Action. Second, we completely determine the family of 2-geodesic transitive circulants.  相似文献   

10.
We study finite four-valent graphs \(\Gamma \) admitting an edge-transitive group G of automorphisms such that G determines and preserves an edge-orientation on \(\Gamma \), and such that at least one G-normal quotient is a cycle (a quotient modulo the orbits of a normal subgroup of G). We show, on the one hand, that the number of distinct cyclic G-normal quotients can be unboundedly large. On the other hand, existence of independent cyclic G-normal quotients (that is, they are not extendable to a common cyclic G-normal quotient) places severe restrictions on the graph \(\Gamma \) and we classify all examples. We show there are five infinite families of such pairs \((\Gamma ,G)\) and in particular that all such graphs have at least one normal quotient which is an unoriented cycle. We compare this new approach with existing treatments for the sub-class of weak metacirculant graphs with these properties, finding that only two infinite families of examples occur in common from both analyses. Several open problems are posed.  相似文献   

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