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1.
Abstract A group G has finite Hirsch-Zaicev rank rhz(G) = r if G has an ascending series whose factors are either infinite cyclic or periodic and if the number of infinite cyclic factors
is exactly r. The authors discuss groups with finite Hirsch-Zaicev rank and the connection between this and groups having finite section
p-rank for some prime p, or p=0. Groups all of whose abelian subgroups are of bounded rank are also discussed.
Keywords: p-rank, locally generalized radical group, Hirsch-Zaicev rank, torsion-free rank, rank
Mathematics Subject Classification (2000): 20F19, 20E25, 20E15 相似文献
2.
Suppose that V is a finite faithful irreducible G-module where G is a finite solvable group of odd order. We prove if the action is quasi-primitive, then either F(G) is abelian or G has at least 212 regular orbits on V. As an application, we prove that when V is a finite faithful completely reducible G-module for a solvable group G of odd order, then there exists v ∈ V such that C G (v) ? F 2(G) (where F 2(G) is the 2nd ascending Fitting subgroup of G). We also generalize a result of Espuelas and Navarro. Let G be a group of odd order and let H be a Hall π-subgroup of G. Let V be a faithful G-module over a finite field of characteristic 2, then there exists v ∈ V such that C H (v) ? O π(G). 相似文献
3.
Emerson de Melo 《代数通讯》2013,41(11):4797-4808
Let M = FH be a finite group that is a product of a normal abelian subgroup F and an abelian subgroup H. Assume that all elements in M?F have prime order p, and F has at most one subgroup of order p. Examples of such groups are dihedral groups for p = 2 and the semidirect product of a cyclic group F by a group H of prime order p such that C F (H) = 1 or |C F (H)| =p and C F/C F (H)(H) = 1. Suppose that M acts on a finite group G in such a manner that C G (F) = 1. We prove that the Fitting height h(G) of G is at most h(C G (H))+ 1. Moreover, the Fitting series of C G (H) coincides with the intersection of C G (H) with the Fitting series of G. 相似文献
4.
Frieder Ladisch 《代数通讯》2013,41(8):2883-2894
We study finite groups G with elements g such that |C G (g)| = |G:G′|. (Such elements generalize fixed-point-free automorphisms of finite groups.) We show that these groups have a unique conjugacy class of nilpotent supplements for the commutator subgroup and, using the classification of finite simple groups, that these groups are solvable. 相似文献
5.
6.
Britta Späth 《Mathematische Zeitschrift》2009,261(3):571-595
Let G be a simply-connected simple algebraic group over an algebraically closed field of characteristic p with a Frobenius map F : G → G and G := G
F
, such that the root system is of exceptional type or G is a Suzuki group or Steinberg’s triality group. We show that all irreducible characters of C
G
(S), the centraliser of S in G, extend to their inertia group in N
G
(S), where S is any F-stable Sylow torus of (G, F). Together with the work in [16] this implies that the McKay conjecture is true for G and odd primes ℓ different from the defining characteristic. Moreover it shows important properties of the associated simple
groups, which are relevant for the proof that the associated simple groups are good in the sense of Isaacs, Malle and Navarro,
as defined in [14].
This research has been supported by the DFG-grant “Die Alperin-McKay-Vermutung für endliche Gruppen” and an Oberwolfach Leibniz
fellowship. 相似文献
7.
《代数通讯》2013,41(12):6135-6147
Abstract Two subgroups H and K of a group G are said to be totally permutable if every subgroup of H permutes with every subgroup of K. In this paper the behaviour of radicals and injectors associated to Fitting classes in a product of pairwise totally permutable finite groups is studied. 相似文献
8.
F. E. A. Johnson 《K-Theory》2005,34(2):141-150
In [F.E.A. Johnson, Stable Modules and the D(2)-Problem, LMS Lecture Notes In Mathematics, vol. 301, CUP (2003)], for finite groups G, we gave a parametrization of the stable class of the augmentation ideal of Z[G] in terms of stably free modules. Whilst the details of this parametrization break down immediately for infinite groups,
nevertheless one may hope to find parallel arguments for restricted classes of infinite groups. Subject to the restriction
that Ext1(Z, Z[G]) = 0, we parametrize the minimal level in Ω1(Z) by means of stably free modules and give a lower estimate for the size of Ω1(Z). 相似文献
9.
On c-Normal Subgroups of Finite Groups 总被引:1,自引:0,他引:1
A subgroup H is said to be c-normal in a group G if there exists a normal subgroup K of G such that G = HK and H K is contained in HG, where HG is the maximal normal subgroup of G. We determine the structures of some groups in which some primary subgroups is c-normal.AMS Mathematics Subject Classification (2000) 20D10 20D20 相似文献
10.
Ali Reza Ashrafi 《Algebra Colloquium》2000,7(2):139-146
For a finite group G, let Cent(G) denote the set of centralizers of single elements of G and #Cent(G) = |Cent(G)|. G is called an n-centralizer group if #Cent(G) = n, and a primitive n-centralizer group if #Cent(G) = #Cent(G/Z(G)) = n. In this paper, we compute #Cent(G) for some finite groups G and prove that, for any positive integer n 2, 3, there exists a finite group G with #Cent(G) = n, which is a question raised by Belcastro and Sherman [2]. We investigate the structure of finite groups G with #Cent(G) = 6 and prove that, if G is a primitive 6-centralizer group, then G/Z(G) A4, the alternating group on four letters. Also, we prove that, if G/Z(G) A4, then #Cent(G) = 6 or 8, and construct a group G with G/Z(G) A4 and #Cent(G) = 8.This research was in part supported by a grant from IPM.2000 Mathematics Subject Classification: 20D99, 20E07 相似文献
11.
Gil Kaplan 《代数通讯》2013,41(5):1722-1729
12.
Raffaele Mosca 《Graphs and Combinatorics》2001,17(3):517-528
Let G be a graph with n vertices, and denote as γ(G) (as θ(G)) the cardinality of a minimum edge cover (of a minimum clique cover) of G. Let E (let C) be the edge-vertex (the clique-vertex) incidence matrix of G; write then P(E)={x∈ℜ
n
:Ex≤1,x≥0}, P(C)={x∈ℜ
n
:Cx≤1,x≥0}, α
E
(G)=max{1
T
x subject to x∈P(E)}, and α
C
(G)= max{1
T
x subject to x∈P(C)}. In this paper we prove that if α
E
(G)=α
C
(G), then γ(G)=θ(G).
Received: May 20, 1998?Final version received: April 12, 1999 相似文献
13.
We classify groups G such that the unit group 𝒰 1(? G) is hypercentral. In the second part, we classify groups G whose modular group algebra has hyperbolic unit groups 𝒰 1(KG). 相似文献
14.
15.
Peter Danchev 《Southeast Asian Bulletin of Mathematics》2003,26(4):559-565
Let R be an arbitrary commutative unitary ring of prime characteristic p and G an arbitrary abelian group whose p-component Gp is an isolated direct sum of torsion-complete abelian groups. Then Gp is a direct factor of S(RG). As a consequence, the same holds when G is a direct sum of groups for which their p-components are torsion-complete groups. In particular when G is p-mixed, it is a direct factor of V(RG) provided R is a field. The formulated results extend a classical theorem of May (Contemp. Math., 1989) for direct sums of cyclic groups and its generalization due to the author (Proc. Amer. Math. Soc., 1997).AMS Subject Classification (2000): Primary 16 U60, 16 S34; Secondary 20 K10, 20 K20, 20 K21. 相似文献
16.
L. Barchini 《Mathematische Annalen》2003,326(2):331-346
Let G be a connected semisimple Lie group contained in its simply connected complexification G
C
. Let KG∩K
C
be a maximal compact subgroup of G. Denote by X
o
the unique closed G-orbit in the full flag manifold ℱ and by 𝒪 the unique open K
C
-orbit in ℱ. The set consisting of the elements gK
C
so that gX
o
⊂𝒪 is an Stein extension of G/K⊂G
C
/K
C
. There is a universal domain , natural form the point of view of group actions which has been conjectured to be Stein. The main result of this paper is
the inclusion . In the second part of the paper I show, under some dominance condition in the parameter, that representations in Dolbeault
cohomology can be realized as holomorphic sections of vector bundles over .
Received: 9 September 2002 / Revised version: 12 July 2002 /
Published online: 8 April 2003
Mathematics Subject Classification (2002): 22E30
Research partially supported by NSF grant DMS-9801605 and DMS 0074991. 相似文献
17.
Thomas Meixner 《Israel Journal of Mathematics》1985,51(1-2):68-78
For a (finite) groupG and some prime powerp
n, theH
p
n
-subgroupH
pn (G) is defined byH
p
n
(G)=〈xεG|x
pn≠1〉. A groupH≠1 is called aH
p
n
-group, if there is a finite groupG such thatH is isomorphic toH
p
n
(G) andH
p
n
(G)≠G. It is known that the Fitting length of a solvableH
p
n
-group cannot be arbitrarily large: Hartley and Rae proved in 1973 that it is bounded by some quadratic function ofn. In the following paper, we show that it is even bounded by some linear function ofn. In view of known examples of solvableH
p
n
-groups having Fitting lengthn, this result is “almost” best possible. 相似文献
18.
19.
If we denote byL(G) the semisimple radical of a groupG, we prove in this paper that ℒ=G|G=CG(L(G))L(G) is a not central normal Fitting class. Moreover, all ( haveN-groups haveN-injectors. 相似文献
20.
On the full automorphism group of a graph 总被引:11,自引:0,他引:11
C. D. Godsil 《Combinatorica》1981,1(3):243-256
While it is easy to characterize the graphs on which a given transitive permutation groupG acts, it is very difficult to characterize the graphsX with Aut (X)=G. We prove here that for the certain transitive permutation groups a simple necessary condition is also sufficient. As a corollary
we find that, whenG is ap-group with no homomorphism ontoZ
p
wrZ
p
, almost all Cayley graphs ofG have automorphism group isomorphic toG. 相似文献