共查询到20条相似文献,搜索用时 31 毫秒
1.
Let G be a group, Aut(G) and L(G) denote the full automorphisms group and absolute centre of G, respectively. The automorphism \({\alpha\in Aut(G)}\) is called autocentral if \({g^{-1}\alpha(g)\in L(G)}\), for all \({g\in G}\). In the present paper, we investigate the properties of such automorphisms. 相似文献
2.
Let G be a finite group. A subgroup H of G is s-permutable in G if H permutes with every Sylow subgroup of G. A subgroup H of G is called an \(\mathcal {SSH}\)-subgroup in G if G has an s-permutable subgroup K such that \(H^{sG} = HK\) and \(H^g \cap N_K (H) \leqslant H\), for all \(g \in G\), where \(H^{sG}\) is the intersection of all s-permutable subgroups of G containing H. We study the structure of finite groups under the assumption that the maximal or the minimal subgroups of Sylow subgroups of some normal subgroups of G are \(\mathcal {SSH}\)-subgroups in G. Several recent results from the literature are improved and generalized. 相似文献
3.
A group G is called capable if there is a group H such that \({G \cong H/Z(H)}\) is isomorphic to the group of inner automorphisms of H. We consider the situation that G is a finite capable p-group for some prime p. Suppose G has rank \({d(G) \ge 2}\) and Frattini class \({c \ge 1}\), which by definition is the length of a shortest central series of G with all factors being elementary abelian. There is up to isomorphism a unique largest p-group \({G_d^c}\) with rank d and Frattini class c, and G is an epimorphic image of \({G_d^c}\). We prove that this \({G_d^c}\) is capable; more precisely, we have \({G_d^c \cong G_d^{c+1}/Z(G_d^{c+1})}\). 相似文献
4.
The group of bisections of groupoids plays an important role in the study of Lie groupoids. In this paper another construction is introduced. Indeed, for a topological groupoid G, the set of all continuous self-maps f on G such that (x, f(x)) is a composable pair for every \(x\in G\), is denoted by \(S_G\). We show that \(S_G\) by a natural binary operation is a monoid. \(S_G(\alpha )\), the group of units in \(S_G\) precisely consists of those \(f\in S_G\) such that the map \(x\mapsto xf(x)\) is a bijection on G. Similar to the group of bisections, \(S_G(\alpha )\) acts on G from the right and on the space of continuous self-maps on G from the left. It is proved that \(S_G(\alpha )\) with the compact- open topology inherited from C(G, G) is a left topological group. For a compact Hausdorff groupoid G it is proved that the group of bisections of \(G^2\) is isomorphic to the group \(S_G(\alpha )\) and the group of transitive bisections of G, \(Bis_T(G)\), is embedded in \(S_G(\alpha )\), where \(G^2\) is the groupoid of all composable pairs. 相似文献
5.
A Banach algebra \(\mathcal {A}\) is called weakly regular if its multiplicative semigroup is E-inversive. We show that for a unimodular group G which admits an integrable unitary representation, \(L^1(G)\) is weakly regular. Moreover for a locally compact Abelian group, \(L^1(G)\) is weakly regular if and only if G is compact; while \(L^1(G)^{**}\) is weakly regular if and only if G is finite. All of our results hold, if we replace \(L^1(G)\) with M(G). 相似文献
6.
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
7.
In this paper, we introduce the probability that a subgroup H of a finite group G can be normal in G, the subgroup normality degree of H in G, as the ratio of the number of all pairs \({(h, g)\in H\times G}\) such that \({h^g\in H}\) by |H||G|. We give some upper and lower bounds for this probability and obtain the upper bound \({\frac{8}{15}}\) for nontrivial subgroups of finite simple groups. In addition, we obtain explicit formulas for subgroup normality degrees of some particular classes of finite groups. 相似文献
8.
Let \(\mathfrak {F}\) be a subgroup-closed saturated formation of finite groups containing all finite nilpotent groups, and let M be a subgroup of a finite group G normalising the \(\mathfrak {F}\)-residual of every non-subnormal subgroup of G. We show that M normalises the \(\mathfrak {F}\)-residual of every subgroup of G. This answers a question posed by Gong and Isaacs (Arch Math 108:1–7, 2017) when \(\mathfrak {F}\) is the formation of all finite supersoluble groups. 相似文献
9.
Mehdi Ghaffarzadeh Mohsen Ghasemi Mark L. Lewis Hung P. Tong-Viet 《Algebras and Representation Theory》2017,20(3):547-567
Given a finite group G, we say that G has property \(\mathcal P_{k}\) if every set of k distinct irreducible character degrees of G is setwise relatively prime. In this paper, we show that if G is a finite nonsolvable group satisfying \(\mathcal P_{4}, \)then G has at most 8 distinct character degrees. Combining with work of D. Benjamin on finite solvable groups, we deduce that a finite group G has at most 9 distinct character degrees if G has property \(\mathcal P_{4}\) and this bound is sharp. 相似文献
10.
For a finite group G, the set of all prime divisors of |G| is denoted by π(G). P. Shumyatsky introduced the following conjecture, which was included in the “Kourovka Notebook” as Question 17.125: a finite group G always contains a pair of conjugate elements a and b such that π(G) = π(〈a, b〉). Denote by \(\mathfrak{Y}\) the class of all finite groups G such that π(H) ≠ π(G) for every maximal subgroup H in G. Shumyatsky’s conjecture is equivalent to the following conjecture: every group from \(\mathfrak{Y}\) is generated by two conjugate elements. Let \(\mathfrak{V}\) be the class of all finite groups in which every maximal subgroup is a Hall subgroup. It is clear that \(\mathfrak{V} \subseteq \mathfrak{Y}\). We prove that every group from \(\mathfrak{V}\) is generated by two conjugate elements. Thus, Shumyatsky’s conjecture is partially supported. In addition, we study some properties of a smallest order counterexample to Shumyatsky’s conjecture. 相似文献
11.
Ping Kang 《Periodica Mathematica Hungarica》2018,76(2):198-206
For a subgroup of a finite group we introduce a new property called weakly c-normal. Suppose that G is a finite group and H is a subgroup of G. H is said to be weakly c-normal in G if there exists a subnormal subgroup K of G such that \(G=HK\) and \(H\cap K\) is s-quasinormally embedded in G. We fix in every non-cyclic Sylow subgroup P of G some subgroup D satisfying \(1<|D|<|P|\) and study the structure of G under the assumption that every subgroup H of P with \(|H|=|D|\) is weakly c-normal in G. Some recent results are generalized and unified. 相似文献
12.
For a group G, write \(g \sim h\) if \(g, h \in G\) have the same order. The set of sizes of the equivalence classes with respect to this relation is called the same-order type of G; thus it is the set with numbers of elements of each order. In this article we prove that a group is isomorphic to the alternating group \(A_5\) if and only if the same-order type of G is \(\{1,pq,4p,8q\}\) with the p and q primes. 相似文献
13.
This paper describes the module categories for a family of generic Hecke algebras, called Yokonuma-type Hecke algebras. Yokonuma-type Hecke algebras specialize both to the group algebras of the complex reflection groups G(r,1,n) and to the convolution algebras of (B \(^{\prime }\),B \(^{\prime }\))-double cosets in the group algebras of finite general linear groups, for certain subgroups B \(^{\prime }\) consisting of upper triangular matrices. In particular, complete sets of inequivalent, irreducible modules for semisimple specializations of Yokonuma-type Hecke algebras are constructed. 相似文献
14.
Let X be a Hausdorff space equipped with a continuous action of a finite group G and a G-stable family of supports \({\Phi}\). Fix a number field F with ring of integers R. We study the class \({\chi = \sum_j (-1)^j [H^j_\Phi (X, \mathcal{E}) \otimes_R F]}\) in the character group of G over F for any flat G-sheaf \({\mathcal{E}}\) of R-modules over X. Under natural cohomological finiteness conditions we give a formula for \({\chi}\) with respect to the basis given by the irreducible characters of G. We discuss applications of our result concerning the cohomology of arithmetic groups. 相似文献
15.
An ordered set \({W = \{w_1, . . . ,w_k\} \subseteqq V (G)}\) of vertices of G is called a resolving set or locating set for G if every vertex is uniquely determined by its vector of distances to the vertices in W. A resolving set of minimum cardinality is called a metric basis for G and this cardinality is the metric dimension or location number of G, denoted by \({\beta(G)}\). 相似文献
16.
A graph G is called claw-o-heavy if every induced claw (\(K_{1,3}\)) of G has two end-vertices with degree sum at least |V(G)|. For a given graph S, G is called S-f-heavy if for every induced subgraph H of G isomorphic to S and every pair of vertices \(u,v\in V(H)\) with \(d_H(u,v)=2,\) there holds \(\max \{d(u),d(v)\}\ge |V(G)|/2.\) In this paper, we prove that every 2-connected claw-o-heavy and \(Z_3\)-f-heavy graph is hamiltonian (with two exceptional graphs), where \(Z_3\) is the graph obtained by identifying one end-vertex of \(P_4\) (a path with 4 vertices) with one vertex of a triangle. This result gives a positive answer to a problem proposed Ning and Zhang (Discrete Math 313:1715–1725, 2013), and also implies two previous theorems of Faudree et al. and Chen et al., respectively. 相似文献
17.
Let G be a p-nilpotent linear group on a finite vector space V of characteristic p. Suppose that |G||V| is odd. Let P be a Sylow p-subgroup of G. We show that there exist vectors \(v_1\) and \(v_2\) in V such that \(C_G(v_1) \cap C_G(v_2)=P\). A striking conjecture of Malle and Navarro offers a simple global criterion for the nilpotence (in the sense of Broué and Puig) of a p-block of a finite group. Our result implies that this conjecture holds for groups of odd order. 相似文献
18.
David Ginzburg 《Israel Journal of Mathematics》2018,226(1):447-474
Let G(r) denote the metaplectic covering group of the linear algebraic group G. In this paper we study conditions on unramified representations of the group G(r) not to have a nonzero Whittaker function. We state a general Conjecture about the possible unramified characters χ such that the unramified subrepresentation of \(Ind_{{B^{\left( r \right)}}}^{{G^{\left( r \right)}}}{X^{\delta _B^{1/2}}}\) will have no nonzero Whittaker function. We prove this Conjecture for the groups GL n ( r) with r ≥ n ? 1, and for the exceptional groups G 2 ( r) when r ≠ 2. 相似文献
19.
Let G be a finite group acting linearly on a vector space V. We consider the linear symmetry groups \({\text {GL}}(Gv)\) of orbits \(Gv\subseteq V\), where the linear symmetry group \({\text {GL}}(S)\) of a subset \(S\subseteq V\) is defined as the set of all linear maps of the linear span of S which permute S. We assume that V is the linear span of at least one orbit Gv. We define a set of generic points in V, which is Zariski open in V, and show that the groups \({\text {GL}}(Gv)\) for v generic are all isomorphic, and isomorphic to a subgroup of every symmetry group \({\text {GL}}(Gw)\) such that V is the linear span of Gw. If the underlying characteristic is zero, “isomorphic” can be replaced by “conjugate in \({\text {GL}}(V)\).” Moreover, in the characteristic zero case, we show how the character of G on V determines this generic symmetry group. We apply our theory to classify all affine symmetry groups of vertex-transitive polytopes, thereby answering a question of Babai (Geom Dedicata 6(3):331–337, 1977. https://doi.org/10.1007/BF02429904). 相似文献
20.
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