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1.
A finite group G is called p
i
-central of height k if every element of order p
i
of G is contained in the k
th
-term ζ
k
(G) of the ascending central series of G. If p is odd, such a group has to be p-nilpotent (Thm. A). Finite p-central p-groups of height p − 2 can be seen as the dual analogue of finite potent p-groups, i.e., for such a finite p-group P the group P/Ω1(P) is also p-central of height p − 2 (Thm. B). In such a group P, the index of P
p
is less than or equal to the order of the subgroup Ω1(P) (Thm. C). If the Sylow p-subgroup P of a finite group G is p-central of height p − 1, p odd, and N
G
(P) is p-nilpotent, then G is also p-nilpotent (Thm. D). Moreover, if G is a p-soluble finite group, p odd, and P ∈ Syl
p
(G) is p-central of height p − 2, then N
G
(P) controls p-fusion in G (Thm. E). It is well-known that the last two properties hold for Swan groups (see [11]). 相似文献
2.
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
3.
Let G be a finite group. For a finite p-group P the subgroup generated by all elements of order p is denoted by Ω1(p). Zhang [5] proved that if P is a Sylow p-subgroup of G, Ω1(P) ≦ Z(P) and N
G
(Z(P)) has a normal p-complement, then G has a normal p-complement. The object of this paper is to generalize this result.
This paper was partly supported by Hungarian National Foundation for Scientific Research Grant # T049841 and T038059. 相似文献
4.
G. R. T. Hendry 《Periodica Mathematica Hungarica》1990,21(3):205-218
A pathP in a graphG is said to beextendable if there exists a pathP’ inG with the same endvertices asP such thatV(P)⊆V (P’) and |V(P’)|=|V(P)|+1. A graphG ispath extendable if every nonhamiltonian path inG is extendable. We investigate the extent to which known sufficient conditions for a graph to be hamiltonian-connected imply
the extendability of paths in the graph. Several theorems are proved: for example, it is shown that ifG is a graph of orderp in which the degree sum of each pair of non-adjacent vertices is at leastp+1 andP is a nonextendable path of orderk inG thenk≤(p+1)/2 and 〈V (P)〉≅K
k
orK
k
−e. As corollaries of this we deduce that if δ(G)≥(p+2)/2 or if the degree sum of each pair of nonadjacent vertices inG is at least (3p−3)/2 thenG is path extendable, which strengthen results of Williamson [13]. 相似文献
5.
Let p(G) and c(G) denote the number of vertices in a longest path and a longest cycle, respectively, of a finite, simple graph G. Define σ4(G)=min{d(x
1)+d(x
2)+ d(x
3)+d(x
4) | {x
1,…,x
4} is independent in G}. In this paper, the difference p(G)−c(G) is considered for 2-connected graphs G with σ4(G)≥|V(G)|+3. Among others, we show that p(G)−c(G)≤2 or every longest path in G is a dominating path.
Received: August 28, 2000 Final version received: May 23, 2002 相似文献
6.
Recognition of the Projective Special Linear Group over GF(3) 总被引:1,自引:0,他引:1
Let P be a finite group and denote by w(P) the set of its element orders. P is called k-recognizable by the set of its element orders if for any finte group G with ω(G) =ω(P) there are, up to isomorphism, k finite groups G such that G ≌P. In this paper we will prove that the group Lp(3), where p 〉 3 is a prime number, is at most 2-recognizable. 相似文献
7.
Let (P,≤) be a partially ordered set. The poset Boolean algebra of P, denoted F(P), is defined as follows: The set of generators of F(P) is {x
p
: p∈P}, and the set of relations is {x
p
⋅x
q
=x
p
: p≤q}. We say that a Boolean algebra B is well-generated, if B has a sublattice G such that G generates B and (G,≤
B
|G) is well-founded. A well-generated algebra is superatomic.
THEOREM 1. Let (P,≤) be a partially ordered set. The following are equivalent. (i) P does not contain an infinite set of pairwise incomparable elements, and P does not contain a subset isomorphic to the chain of rational numbers, (ii) F(P) is superatomic, (iii) F(P) is well-generated.
The equivalence (i) ⇔ (ii) is due to M. Pouzet. A partially ordered set W is well-ordered, if W does not contain a strictly decreasing infinite sequence, and W does not contain an infinite set of pairwise incomparable elements.
THEOREM 2. Let F(P) be a superatomic poset algebra. Then there are a well-ordered set W and a subalgebra B of F(W), such that F(P) is a homomorphic image of B.
This is similar but weaker than the fact that every interval algebra of a scattered chain is embeddable in an ordinal algebra.
Remember that an interval algebra is a special case of a poset algebra.
This revised version was published online in June 2006 with corrections to the Cover Date. 相似文献
8.
John Shareshian 《Journal of Algebra》1998,210(2):195
LetGbe a finite group, and define the function[formula]where μ is the Möbius function on the subgroup lattice ofG. The functionP(G, s) is the multiplicative inverse of a zeta function forG, as described by Mann and Boston. Boston conjectured thatP′(G, 1) = 0 ifGis a nonabelian simple. We will prove a generalization of this conjecture, showing thatP′(G, 1) = 0 unlessG/Op(G) is cyclic for some primep. 相似文献
9.
In this paper it is proved that ifp is a prime dividing the order of a groupG with (|G|,p − 1) = 1 andP a Sylowp-subgroup ofG, thenG isp-nilpotent if every subgroup ofP ∩G
N
of orderp is permutable inN
G
(P) and whenp = 2 either every cyclic subgroup ofP ∩G
N
of order 4 is permutable inN
G
(P) orP is quaternion-free. Some applications of this result are given.
The research of the first author is supported by a grant of Shanxi University and a research grant of Shanxi Province, PR
China.
The research of the second author is partially supported by a UGC(HK) grant #2160126 (1999/2000). 相似文献
10.
Let G be a finite group. The prime graph Γ(G) of G is defined as follows. The vertices of Γ(G) are the primes dividing the order of G and two distinct vertices p and p′ are joined by an edge if there is an element in G of order pp′. We denote by k(Γ(G)) the number of isomorphism classes of finite groups H satisfying Γ(G) = Γ(H). Given a natural number r, a finite group G is called r-recognizable by prime graph if k(Γ(G)) = r. In Shen et al. (Sib. Math. J. 51(2):244–254, 2010), it is proved that if p is an odd prime, then B
p
(3) is recognizable by element orders. In this paper as the main result, we show that if G is a finite group such that Γ(G) = Γ(B
p
(3)), where p > 3 is an odd prime, then G @ Bp(3){G\cong B_p(3)} or C
p
(3). Also if Γ(G) = Γ(B
3(3)), then G @ B3(3), C3(3), D4(3){G\cong B_3(3), C_3(3), D_4(3)}, or G/O2(G) @ Aut(2B2(8)){G/O_2(G)\cong {\rm Aut}(^2B_2(8))}. As a corollary, the main result of the above paper is obtained. 相似文献
11.
Jochen Koenigsmann 《manuscripta mathematica》1998,95(2):251-271
Let p be a prime > 2, let F be a field of characteristic ≠p containing a primitive p-th root of unity and let G
F
(p) be the Galois group of the maximal Galois-p-extension of F. If rk G
F
(p)≤ 4 then G
F
(p) is a free pro-p product of metabelian groups or G
F
(p) is a Demuškin group of rank 4.
Received: 3 September 1997 / Revised version: 3 October 1997 相似文献
12.
Keiko Kotani 《Graphs and Combinatorics》2001,17(3):511-515
Let G be a connected graph without loops and without multiple edges, and let p be an integer such that 0 < p<|V(G)|. Let f be an integer-valued function on V(G) such that 2≤f(x)≤ deg
G
(x) for all x∈V(G). We show that if every connected induced subgraph of order p of G has an f-factor, then G has an f-factor, unless ∑
x
∈
V
(
G
)
f(x) is odd.
Received: June 29, 1998?Final version received: July 30, 1999 相似文献
13.
Pablo Centella 《代数通讯》2013,41(1):312-321
Let G be a finite solvable group and let p be a prime. Let P ∈ Syl p (G) and N = N G (P). We prove that there exists a natural bijection between the 2-Brauer irreducible characters of p′-degree of G and those of N G (P). 相似文献
14.
In representation theory of finite groups, one of the most important and interesting problems is that, for a p-block A of a finite group G where p is a prime, the numbers k(A) and ℓ(A) of irreducible ordinary and Brauer characters, respectively, of G in A are p-locally determined. We calculate k(A) and ℓ(A) for the cases where A is a full defect p-block of G, namely, a defect group P of A is a Sylow p-subgroup of G and P is a nonabelian metacyclic p-group M
n+1(p) of order p
n+1 and exponent p
n
for
n \geqslant 2{n \geqslant 2}, and where A is not necessarily a full defect p-block but its defect group P = M
n+1(p) is normal in G. The proof is independent of the classification of finite simple groups. 相似文献
15.
For a graph G, let diff(G) = p(G) − c(G), where p(G) and c(G) denote the orders of a longest path and a longest cycle in G, respectively. Let G be a 3-connected graph of order n. In the paper, we give a best-possible lower bound to σ4(G) to assure diff(G) ≤ 1. The result settles a conjecture in J. Graph Theory 37 (2001), 137–156. 相似文献
16.
In 1963, Vizing [Vichysl.Sistemy 9 (1963), 30–43] conjectured that γ(G × H) ≥ γ(G)γ(H), where G × H denotes the cartesian product of graphs, and γ(G) is the domination number. In this paper we define the extraction number x(G) and we prove that P2(G) ≤ x(G), and γ(G × H) ≥ x(G)γ(H), where P2(G) is the 2-packing number of G. Though the equality x(G) = γ(G) is proven to hold in several classes of graphs, we construct an infinite family of graphs which do not satisfy this condition. Also, we show the following lower bound: γ(G × H) ≥ γ(G)P2(H) + P2(G)(γ(H) − P2(H)). © 1996 John Wiley & Sons, Inc. 相似文献
17.
In this paper, a finite group G with IAut(G) : P(G)I ~- p or pq is determined, where P(G) is the power automorphism group of G, and p, q are distinct primes. Especially, we prove that a finite group G satisfies |Aut(G) : P(G)|= pq if and only if Aut(G)/P(G) ≌S3. Also, some other classes of finite groups are investigated and classified, which are necessary for the proof of our main results. 相似文献
18.
Schur multiplicators of infinite pro-<Emphasis Type="Italic">p</Emphasis>-groups with finite coclass
Bettina Eick 《Israel Journal of Mathematics》2008,166(1):147-156
Let G be an infinite pro-p-group of finite coclass and let M(G) be its Schur multiplicator. For p > 2, we determine the isomorphism type of Hom(M(G), ℤp), where ℤp denotes the p-adic integers, and show that M(G) is infinite. For p = 2, we investigate the Schur multiplicators of the infinite pro-2-groups of small coclass and show that M(G) can be infinite, finite or even trivial. 相似文献
19.
Let G be a non-abelian group and associate a non-commuting graph ∇(G) with G as follows: the vertex set of ∇(G) is G\Z(G) with two vertices x and y joined by an edge whenever the commutator of x and y is not the identity. In this short paper we prove that if G is a finite group with ∇(G) ≅ ∇(M), where M = L
2(q) (q = p
n
, p is a prime), then G ≅ M.
相似文献
20.
Given 1 ≤ p < ∞, a compact abelian group G and a p-multiplier ${\psi : \Gamma \to {\mathbb C}}Given 1 ≤ p < ∞, a compact abelian group G and a p-multiplier
y: G? \mathbb C{\psi : \Gamma \to {\mathbb C}} (with Γ the dual group), we study the optimal domain of the multiplier operator T(p)y : Lp (G) ? Lp (G){T^{(p)}_\psi : L^p (G) \to L^p (G)}. This is the largest Banach function space, denoted by L1(m(p)y){L^1(m^{(p)}_\psi)}, with order continuous norm into which L
p
(G) is embedded and to which T(p)y{ T^{(p)}_\psi} has a continuous L
p
(G)-valued extension. Compactness conditions for the optimal extension are given, as well as criteria for those ψ for which L1(m(p)y) = Lp (G){L^1(m^{(p)}_\psi) = L^p (G)} is as small as possible and also for those ψ for which L1(m(p)y) = L1 (G){L^1(m^{(p)}_\psi) = L^1 (G)} is as large as possible. Several results and examples are presented for cases when
Lp (G) \subsetneqq L1(m(p)y) \subsetneqq L1 (G){L^p (G) \subsetneqq L^1(m^{(p)}_\psi) \subsetneqq L^1 (G)}. 相似文献