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1.
Wenbin Guo 《manuscripta mathematica》2008,127(2):139-150
Let G be a finite group and a formation of finite groups. We say that a subgroup H of G is -supplemented in G if there exists a subgroup T of G such that G = TH and is contained in the -hypercenter of G/H
G
. In this paper, we use -supplemented subgroups to study the structure of finite groups. A series of previously known results are unified and generalized.
Research of the author is supported by a NNSF grant of China (Grant #10771180). 相似文献
2.
Finite groups with minimal 1-PIM 总被引:1,自引:0,他引:1
Let be a field of characteristic and let G be a finite group. It is well-known that the dimension of the minimal projective cover (the so-called 1-PIM) of the trivial left -module is a multiple of the -part of the order of G. In this note we study finite groups G satisfying . In particular, we classify the non-abelian finite simple groups G and primes satisfying this identity (Theorem A). As a consequence we show that finite soluble groups are precisely those finite groups
which satisfy this identity for all prime numbers (Corollary B). Another consequence is the fact that the validity of this identity for a finite group G and for a small prime number implies the existence of an -Hall subgroup for G (Theorem C). An important tool in our proofs is the super-multiplicativity of the dimension of the 1-PIM over short exact
sequences (Proposition 2.2). 相似文献
3.
Jan Florek 《Algebra Universalis》2008,58(3):341-347
On a partially ordered set G the orthogonality relation is defined by incomparability and is a complete orthocomplemented lattice of double orthoclosed sets. We will prove that the atom space of the lattice has the same order structure as G. Thus if G is a partially ordered set (an ordered group, or an ordered vector space), then is a canonically partially ordered set (an ordered quotient group, or an ordered quotient vector space, respectively). We
will also prove: if G is an ordered group with a positive cone P, then the lattice has the covering property iff , where g is an element of G and M is the intersection of all maximal subgroups contained in .
Received August 1, 2006; accepted in final form May 29, 2007. 相似文献
4.
Volker Runde 《Archiv der Mathematik》2009,92(5):525-530
We show that the biflatness—in the sense of A. Ya. Helemskiĭ—of the Fourier algebra A(G) of a locally compact group G forces G to either have an abelian subgroup of finite index or to be non-amenable without containing as a closed subgroup. An analogous dichotomy is obtained for biprojectivity.
Received: 4 August 2008 相似文献
5.
In this paper, we get the main theorem: Let p be a prime dividing the order of G and , where and H is p
′-Hall subgroup of G. Let δ be a complete set of Sylow subgroups of H. If G satisfies the following conditions: i) is a p-group; ii) for any maximal M of P, M is δ-permutable in H, then G is p-nilpotent. Some known results are generalized.
Received: 12 September 2007, Revised: 28 February 2008 相似文献
6.
Let G be a finite p-solvable group. Let P ∈ Syl
p
(G) and N = N
G
(P). We prove that there exists a natural bijection between the irreducible constituents of p′-degree of the principal projective character of G and those of .
Received: 2 May 2007, Revised: 17 September 2007 相似文献
7.
For a Hall system of a finite solvable group G, it is known that the set
of -permutable subgroups is a sublattice of the subgroup lattice of G. We investigate the class SPM of groups in which the lattice
is modular. We prove that if
is modular, then U V for all
(an evidently stronger condition). Both of these phenomena—the modularity of
and whether two -permutable subgroups U and V permute with each other—are shown to be determined locally, by what happens at each prime. The class SPM is shown to be quotient closed, but not direct product or subgroup closed.This revised electronic version of the Abstract includes the formulas that were missing in the previous electronic version published online in September 2004. 相似文献
8.
Fuquan Fang 《Mathematische Zeitschrift》2008,259(3):643-656
Let M be an oriented compact Riemannian 4-manifold with positive sectional curvature. Let G be a finite subgroup of the isometry group of M. We prove that, if G is a finite group of order , then
Moreover, M is homeomorphic to if G is non-abelian of odd order.
Supported partially by NSF Grant 19925104 of China, 973 project of Foundation Science of China and the Max-Planck Institut
für Mathematik at Bonn. 相似文献
(i) | G is isomorphic to a subgroup of PU(3) if |G| is odd; |
(ii) | G contains an index at most 2 normal subgroup which is isomorphic to a subgroup of SO(5) or PU(3) if |G| is even, and M is not homeomorphic to S 4. |
9.
John K. McVey 《Archiv der Mathematik》2006,87(2):97-103
When G is a finite nonabelian group, we associate the common-divisor graph with G by letting nontrivial degrees in cd(G) = {χ(1) | χ∈Irr(G)} be the vertices and making distinct vertices adjacent if they have a common nontrivial divisor. A set
of vertices for this graph is said to be strongly connective for cd(G) if there is some prime which divides every member of
, and every vertex outside of
is adjacent to some member of
. When G is nonsolvable, we provide sufficiency conditions for cd(G) to have a strongly connective subset. We also extend a previously known result about groups with nonabelian solvable quotients,
and prove for arbitrary groups G that if the associated graph is connected and has a diameter bounded by 2, then indeed cd(G) has a strongly connective subset. The major focus is on when the derived subgroup G′ is perfect.
Received: 23 July 2005 相似文献
10.
Hatem Hamrouni 《manuscripta mathematica》2008,127(4):511-519
Let G be a connected and simply connected nilpotent Lie group and A a closed connected subgroup of G. Let Γ be a discrete cocompact subgroup of G. In the first part of this paper we give the direct integral decomposition of the up–down representation . As a consequence, we establish a necessary and sufficient condition for A to act ergodically on G/Γ in the case when Γ is a lattice subgroup of G and A is a one-parameter subgroup of G. 相似文献
11.
Martin R. Pettet 《Archiv der Mathematik》2008,91(4):289-299
Let G be an A-group (i.e. a group in which xx
α
= x
α
x for all and let denote the subgroup of Aut(G) consisting of all automorphisms that leave invariant the centralizer of each element of G. The quotient is an elementary abelian 2-group and natural analogies exist to suggest that it might always be trivial. It is shown that,
in fact, for any odd prime p and any positive integer r, there exist infinitely many finite pA-groups G for which has rank r.
Received: 23 March 2008, Revised: 20 May 2008 相似文献
12.
Assume that we have a (compact) Riemann surface S, of genus greater than 2, with , where is the complex unit disc and Γ is a surface Fuchsian group. Let us further consider that S has an automorphism group G in such a way that the orbifold S/G is isomorphic to where is a Fuchsian group such that and has signature σ appearing in the list of non-finitely maximal signatures of Fuchsian groups of Theorems 1 and 2 in [6]. We
establish an algebraic condition for G such that if G satisfies such a condition then the group of automorphisms of S is strictly greater than G, i.e., the surface S is more symmetric that we are supposing. In these cases, we establish analytic information on S from topological and algebraic conditions.
Received: 4 April 2008 相似文献
13.
Let
be a limit group,
a non-trivial subgroup, and N the normaliser of S. If
has finite
-dimension, then S is finitely generated and either N/S is finite or N is abelian. This result has applications to the study of subdirect products of limit groups.Supported in part by Franco-British Alliance project PN 05.004. 相似文献
14.
Olympia Talelli 《Archiv der Mathematik》2007,89(1):24-32
We define a group G to be of type Φ if it has the property that for every
-module G, proj.
G < ∞ iff proj.
H G < ∞ for every finite subgroup H of G. We conjecture that the type Φ is an algebraic characterization of those groups G which admit a finite dimensional model for
, the classifying space for the family of the finite subgroups of G. We also conjecture that the type Φ is equivalent to spli being finite, where spli
is the supremum of the projective lengths of the injective
-modules. Here we prove certain parts of these conjectures.
The project is cofounded by the European Social Fund and National Resources–EPEAK II–Pythagoras.
Received: 21 June 2006 相似文献
15.
We determine the Zariski-dense subgroups of Chevalley groups and their twisted analogues over infinite algebraic extensions
of finite fields. It turns out that these are essentially forms of the same group (possibly becoming twisted) over smaller
infinite fields. It follows from our classification that if
is a simple algebraic group over the algebraic closure of a finite field, then a dense subgroup of
can never be maximal, and so the maximal subgroups of
are necessarily closed. It follows that Seitz’s determination of the closed maximal subgroups of
actually gives all the maximal subgroups.
It also enables us to prove that ifG is a simple Chevalley group or twisted type over an infinite algebraic extension of a finite field, then in every non-trivial
permutation representation ofG, every finite subgroup has a regular orbit. It follows that every non-trivial permutation module forG over a fieldk iskG-faithful. This is relevant to a programme of studying ideals in group rings of simple locally finite groups.
To John Thompson in recognition of his many outstanding contributions to group theory 相似文献
16.
Mehdi Shabani Attar 《Archiv der Mathematik》2007,89(4):296-297
Let G be a group and let Aut
c
(G) be the group of central automorphisms of G. Let
be the set of all central automorphisms of G fixing Z(G) elementwise. In this paper we prove that if G is a finite p-group, then
= Inn(G) if and only if G is abelian or G is nilpotent of class 2 and Z(G) is cyclic.
This work was supported in part by the Center of Excellence for Mathematics, University of Isfahan, Iran.
Received: 30 October 2006 相似文献
17.
In this note, we show that if
is a π-partial character of the π-separable group
is a chain of normal subgroups of G, and H is a Hall π-subgroup of G, then
has a Fong character α
Irr(H) such that for every subgroup
, every irreducible constituent of α
H∩N
is Fong for N. We also show that if
is quasi-primitive, then for every normal subgroup M of G the irreducible constituents of
are Fong for M.
Received: 21 July 2006 Revised: 17 January 2007 相似文献
18.
Alexandre Turull 《Archiv der Mathematik》2005,84(2):97-106
Let G be a finite group, N a normal subgroup of G, and an irreducible character of G. Clifford Theory studies a whole collection of related irreducible characters of all the subgroups of G that contain N. The relationships among these characters as well as their Schur indices are controlled by the Clifford class c Clif(G/N, F) of with respect to N over some field F. This is an equivalence class of central simple G/N-algebras. Assume now that G/N is cyclic. One can obtain a new isoclinic group
and character
by multiplying each element of each coset of N in G by an appropriate power of a fixed root of unity . We show that there is a simple formula to calculate the Clifford class
of
in terms of c and . Hence, the Clifford class c controls not only the Schur index of the characters of all the subgroups of G that contain N, it also controls the Schur indices of the characters of the corresponding characters of the isoclinic groups
When is a |G/N|-th root of 1, our formula shows that then
When = i and |G/N| = 2, the implicit transformation on Clif(Z/2Z, F) yields a group homomorphism of the group structure introduced on the Brauer-Wall group of F to describe the Schur indices of all the irreducible characters of the double covers of the symmetric and alternating groups.Received: 17 August 2001 相似文献
19.
Marja Kankaanrinta 《Geometriae Dedicata》2008,136(1):167-173
In this paper we prove an equivariant version of the uniformization theorem for closed subanalytic sets: Let G be a Lie group and let M be a proper real analytic G-manifold. Let X be a closed subanalytic G-invariant subset of M. We show that there exist a proper real analytic G-manifold N of the same dimension as X and a proper real analytic G-equivariant map such that .
相似文献
20.
We improve Margulis lemma for a compact connected Lie group G: there is a neighborhood U of the identity such that for any finite subgroup , generates an abelian group. We show that for each n, there exists an integer , such that if H is a closed subgroup of a compact connected Lie group G of dimension n, then the quotient group, H/H
0, has an abelian subgroup of index , where H
0 is the identity component of H. As an application, we show that the fundamental group of the homogeneous space G/H has an abelian subgroup of index . We show this same property for the fundamental groups of almost non-negatively curved n-manifolds whose universal coverings are not collapsed.
X. Rong: supported partially by NSF Grant DMS 0504534 and by a reach found from Beijing Normal University.
Y. Wang: supported partially by LMAM of Peking University and by NSFC 10671018. 相似文献