共查询到20条相似文献,搜索用时 15 毫秒
1.
A subgroup H of a finite group G is said to be complemented in G if there exists a subgroup K of G such that G=HK and H∩K=1. In this paper, it is proved that a finite group G is p-nilpotent provided p is the smallest prime number dividing the order of G and every minimal subgroup of the p-focal subgroup of G is complemented in NG(P), where P is a Sylow p-subgroup of G. As some applications, some interesting results related with complemented minimal subgroups of focal subgroups are obtained. 相似文献
2.
A new condition for solvable groups 总被引:1,自引:0,他引:1
A subgroup H of G is called complemented in G if there exists a subgroup K of G such that and . The aim of this paper is to prove the following: A finite group G is solvable if and only if its Sylow 3-, 5- and 7-subgroups are complemented in G. 相似文献
3.
Let 𝔉 be a class of groups and G a finite group. A maximal subgroup M of G is called 𝔉-abnormal provided G∕MG?𝔉. Let K<H be subgroup of G. Then we say that (K,H) is an 𝔉-abnormal pair of G provided K is a maximal 𝔉-abnormal subgroup of H. Let A be a subgroup of G. Then we say that A is 𝔉-quasipermutable in G provided A either covers or avoids every 𝔉-abnormal pair of G. In this paper, we consider some applications of 𝔉-quasipermutable subgroups. 相似文献
4.
A subgroup H of G is said to be $\pi$-quasinormal in G if it
permute with every Sylow subgroup of G.
In this paper, we extend the study on the structure of a finite group under the
assumption that some subgroups of G are
$\pi$-quasinormal in G. The main result we proved
in this paper is the following:Theorem 3.4.
Let ${\cal F}$ be a saturated formation
containing the supersolvable groups. Suppose that G is a group with a
normal subgroup H such that $G/H \in {\cal F}$,
and all maximal subgroups of any Sylow subgroup of $F^{*}(H)$ are $\pi$-quasinormal in
G, then $G \in {\cal F}$.
Received: 10 May 2002 相似文献
5.
M. J. Curran 《Archiv der Mathematik》2008,90(6):483-489
This note considers a finite group G = HK, which is a product of a subgroup H and a normal subgroup K, and determines subgroups of Aut G. The special case when G is a nonsplit metacyclic p-group, where p is odd, is then considered and the structure of its automorphism group Aut G is given.
Received: 13 September 2007, Revised: 22 November 2007 相似文献
6.
A subgroup H of a finite group G is said to have the semi cover-avoiding property in G if there is a normal series of G such that H covers or avoids every normal factor of the series. In this paper, some new results are obtained based on the assumption that some subgroups have the semi cover-avoiding property in the group. 相似文献
7.
Finite groups G=AB factorized by two subgroups A and B such that every subgroup of A permutes with every subgroup of B are studied in this paper. The behaviour of such products with respect to the class of finite groups in which Sylow-permutability is transitive is analyzed. 相似文献
8.
We prove conditions for a product of distinct subgroups of an arbitrary group G to be a subgroup of G. In particular, the normal closure of any A ≤ G is equal to the product of some distinct conjugates of A. As an application of the later result we derive constraints on the size of a nontrivial conjugacy class of a finite non-Abelian simple group. 相似文献
10.
Helge Glöckner 《manuscripta mathematica》1998,97(2):205-215
Let G be a p-adic Lie group. Then G is a locally compact, totally disconnected group, to which Willis [14] associates its scale function G : G→ℕ. We show that s can be computed on the Lie algebra level. The image of s consists of powers of p. If G is a linear algebraic group over ℚ
p
, s(x)=s(h) is determined by the semisimple part h of x∈G. For every finite extension K of ℚ
p
, the scale functions of G and H:=G(K) are related by s
H
∣
G
=s
G
[
K
:ℚ
p
]. More generally, we clarify the relations between the scale function of
a p-adic Lie group and the scale functions of its closed subgroups and Hausdorff quotients.
Received: 20 February 1997; Revised version: 18 May 1998 相似文献
11.
In this paper, it is proved that a finite group G is p-nilpotent if every minimal subgroup of P∩Op(G) is permutable in P and NG(P) is p-nilpotent, and when p=2 either [Ω2(P∩Op(G)),P]Ω1(P∩Op(G)) or P is quaternion-free, where p is a prime dividing the order of G and P is a Sylow p-subgroup of G. By using this result, we may get a series of corollaries for p-nilpotence, which contain some known results. Some other applications of this result are also given. 相似文献
12.
Let KG be a group algebra of a finite p-group G over a finite field Kof characteristic p. We compute the order of the unitary subgroup of the group of units when G is either an extraspecial 2-group or the central product of such a group with a cyclic group of order 4 or G has an abelian subgroup A of index 2 and an element b such that b inverts each element of A. 相似文献
13.
It is known that in a finite solvable group G, a subgroup H is abnormal if and only if every subgroup of G containing H is self-normalizing in G. Although, in general, the assumption of solvability cannot be dropped, in this paper we prove the theorem for the special case G = An and H a second maximal intransitive subgroup of An.Received: 1 July 2003 相似文献
14.
Let H and K be spherical subgroups of a reductive complex group G. In many cases, detailed knowledge of the double coset space H\G/K is of fundamental importance in group theory and representation theory. If H or K is parabolic, then H\G/K is finite, and we recall the classification of the double cosets in several important cases. If H=K is a symmetric subgroup of G, then the double coset space K\G/K (and the corresponding invariant theoretic quotient) are no longer finite, but several nice properties hold, including an analogue of the Chevalley restriction theorem. These properties were generalized by Helminck and Schwarz (Duke Math. J.
106(2) (2001), pp. 237–279) to the case where H and K are fixed point groups of commuting involutions. We recall Helminck and Schwarz's main results. We also give examples to show the difficulty in extending these results if we allow H=K to be a reductive spherical (nonsymmetric) subgroup or if we have H symmetric and K spherical reductive. 相似文献
15.
A subgroup H of a finite group G is called c*-supplemented in G if there exists a subgroup K of G such that G = HK and H ∩ K is S-quasinormally embedded in G. In this paper, we investigate the local c*-supplementation of maximal subgroups of some Sylow p-subgroup and present some sufficient and necessary conditions for a finite group to be p-nilpotent. As applications, we give some sufficient conditions for a finite group to be in a saturated formation. 相似文献
16.
We show that the existence of a non-metrizable compact subspace of a topological group G often implies that G contains an uncountable supersequence (a copy of the one-point compactification of an uncountable discrete space). The existence of uncountable supersequences in a topological group has a strong impact on bounded subsets of the group. For example, if a topological group G contains an uncountable supersequence and K is a closed bounded subset of G which does not contain uncountable supersequences, then any subset A of K is bounded in G?(K?A). We also show that every precompact Abelian topological group H can be embedded as a closed subgroup into a precompact Abelian topological group G such that H is bounded in G and all bounded subsets of the quotient group G/H are finite. This complements Ursul's result on closed embeddings of precompact groups to pseudocompact groups. 相似文献
17.
In a finite group G every element can be factorized in such a way that there is one factor for each prime divisor p of | G |, and the order of this factor is pα for some integer α ≧ 0. We define g ∈G to be uniquely factorizable if it has just one such factorization (whose factors must be pairwise commuting). We consider the existence of uniquely factorizable
elements and its relation to the solvability of the group. We prove that G is solvable if and only if the set of all uniquely factorizable elements of G is the Fitting subgroup of G. We also prove various sufficient conditions for the non-existence of uniquely factorizable elements in non-solvable groups.
Received: 9 June 2005 相似文献
18.
Thomas P. Wakefield 《Algebras and Representation Theory》2012,15(3):427-448
Let G denote a finite group and cd (G) the set of irreducible character degrees of G. Bertram Huppert conjectured that if H is a finite nonabelian simple group such that cd (G) = cd (H), then G ≅ H × A, where A is an abelian group. Huppert verified the conjecture for PSp4(q) when q = 3, 4, 5, or 7. In this paper, we extend Huppert’s results and verify the conjecture for PSp4(q) for all q. This demonstrates progress toward the goal of verifying the conjecture for all nonabelian simple groups of Lie type of rank
two. 相似文献
19.
Automorphisms of direct products of finite groups 总被引:1,自引:0,他引:1
This paper shows that if H and K are finite groups with no common direct factor and G = H × K, then the structure and order of Aut G can be simply expressed in terms of Aut H, Aut K and the central homomorphism groups
Hom (H, Z(K)) and Hom (K, Z(H)).
Received: 18 April 2005; revised: 9 June 2005 相似文献
20.
A subgroup H of a finite group
G is called c-normal in
G if there exists a normal subgroup
N of G such that
G = HN and $H \cap N \leq H_{G} = {\rm core}_{G}(H)$. In this paper, we investigate the class of groups
of which every maximal subgroup of its Sylow
p-subgroup is c-normal and the
class of groups of which some minimal subgroups of its Sylow
p-subgroup is c-normal for some prime number
p. Some interesting results are obtained and
consequently, many known results related to
p-nilpotent groups and
p-supersolvable groups are generalized. 相似文献