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1.
Let G be a finite group. A subgroup K of a group G is called an ?-subgroup of G if N G (K) ∩ K x ≦ K for all x ? G. The set of all ?-subgroups of G will be denoted by ?(G). Let P be a nontrivial p-group. A chain of subgroups 1 = P 0 ? P 1 ? ··· ? P n = P is called a maximal chain of P provided that |P i : P i?1| = p, i = 1, 2, ···, n. A nontrivial p-subgroup P of G is called weakly supersolvably embedded in G if P has a maximal chain 1 = P 0 ? P 1 ? ··· ? P i ? ··· ? P n = P such that P i ? ?(G) for i = 1, 2, ···, n. Using the concept of weakly supersolvably embedded, we obtain new characterizations of p-nilpotent and supersolvable finite groups. 相似文献
2.
Let G be a non-abelian group and Z(G) be the center of G. Associate a graph Γ G (called noncommuting graph of G) with G as follows: Take G?Z(G) as the vertices of Γ G , and join two distinct vertices x and y, whenever xy ≠ yx. Here, we prove that “the commutativity pattern of a finite non-abelian p-group determine its order among the class of groups"; this means that if P is a finite non-abelian p-group such that Γ P ? Γ H for some group H, then |P| = |H|. 相似文献
3.
Let T be a complete local (Noetherian) ring with maximal ideal M, P a nonmaximal ideal of T, and C = {Q 1, Q 2,…} a (nonempty) finite or countable set of nonmaximal prime ideals of T. Let {p 1, p 2,…} be a set of nonzero regular elements of T, whose cardinality is the same as that of C. Suppose that p i ∈ Q j if and only if i = j. We give conditions that ensure there is an excellent local unique factorization domain A such that A is a subring of T, the maximal ideal of A is M ∩ A, the (M ∩ A)-adic completion of A is T, and so that the following three conditions hold: (1) p i ∈ A for every i; (2) A ∩ P = (0), and if J is a prime ideal of T with J ∩ A = (0), then J ? P or J ? Q i for some i; (3) for each i, p i A is a prime ideal of A, Q i ∩ A = p i A, and if J is a prime ideal of T with J ? Q i , then J ∩ A ≠ p i A. 相似文献
4.
Lingli Wang 《代数通讯》2013,41(2):523-528
Let G be a nonabelian group and associate a noncommuting 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 1987, Professor J. G. Thompson gave the following conjecture. Thompson's Conjecture. If G is a finite group with Z(G) = 1 and M is a nonabelian simple group satisfying N(G) = N(M), then G ? M, where N(G):={n ∈ ? | G has a conjugacy class of size n}. In 2006, A. Abdollahi, S. Akbari, and H. R. Maimani put forward a conjecture (AAM's conjecture) in Abdollahi et al. (2006) as follows. AAM's Conjecture. Let M be a finite nonabelian simple group and G a group such that ?(G) ? ? (M). Then G ? M. In this short article we prove that if G is a finite group with ?(G) ? ? (A 10), then G ? A 10, where A 10 is the alternating group of degree 10. 相似文献
5.
Let G be a finite group, and let p 1,…, p m be the distinct prime divisors of |G|. Given a sequence 𝒫 =P 1,…, P m , where P i is a Sylow p i -subgroup of G, and g ∈ G, denote by m 𝒫(g) the number of factorizations g = g 1…g m such that g i ∈ P i . Previously, it was shown that the properly normalized average of m 𝒫 over all 𝒫 is a complex character over G termed the Average Sylow Multiplicity Character. The present article identifies the kernel of this character as the subgroup of G consisting of all g ∈ G such that m 𝒫(gx) = m 𝒫(x) for all 𝒫 and all x ∈ G. This result implies a close connection between the kernel and the solvable radical of G. 相似文献
6.
Hirotaka Koga 《代数通讯》2013,41(7):2417-2429
Let R be a commutative noetherian ring and A a noetherian R-algebra. Let P ? ∈ 𝒦b(𝒫 A ) with Hom𝒦(Mod-A)(P ?, P ?[i]) = 0 for i > 0. We will provide a sufficient condition for P ? to be a direct summand of a silting complex. Also, in case Hom𝒦(Mod-A)(P ?, P ?[i]) = 0 for i ≠ 0, we will provide a sufficient condition for P ? to be a direct summand of a tilting complex. 相似文献
7.
Let (S, 𝔫) be an s-dimensional regular local ring with s > 2, and let I = (f, g) be an ideal in S generated by a regular sequence f, g of length two. As in [2, 3], we examine the leading form ideal I* of I in the associated graded ring G: = gr𝔫(S). Let μ G (I*) = n ≥ 3, and let {ξ1, ξ2,…, ξ n } be a minimal homogeneous system of generators of I* such that ξ1 = f* and ξ2 = g*, and c i : = deg ξ i ≤ deg ξ i+1: = c i+1 for each i ≤ n ? 1. For m ≤ n, we say that K m : = (ξ1,…, ξ m )G is an ideal generated by part of a minimal homogeneous generating set of I*. Let D i : = GCD(ξ1,…, ξ i ) and d i = deg D i for i with 1 ≤ i ≤ m. Let K m be perfect with ht G K m = 2. We prove that the following are equivalent: 1. deg ξ i+1 = deg ξ i + (d i?1 ? d i ) +1, for all i with 3 ≤ i ≤ m ? 1; 2. deg ξ i+1 ≤ deg ξ i + (d i?1 ? d i ) +1, for all i with 3 ≤ i ≤ m ? 1. Furthermore, if these equivalent conditions hold, then K m = I*. Moreover, if e(G/K m ) = e(G/I*), we prove that K m = I*. We illustrate with several examples in the cases where K m is or is not perfect. 相似文献
8.
Xianglin Du 《代数通讯》2013,41(4):1345-1359
ABSTRACT Let k(G) be the number of conjugacy classes of finite groups G and π e (G) be the set of the orders of elements in G. Then there exists a non-negative integer k such that k(G) = |π e (G)| + k. We call such groups to be co(k) groups. This article classifies all finite co(1) groups. They are isomorphic to one of the following groups: A 5, L 2(7), S 5, Z 3, Z 4, S 4, A 4, D 10, Hol(Z 5), or Z 3 ? Z 4. 相似文献
9.
Let (R, m) be a Cohen–Macaulay local ring, and let ? = {F i } i∈? be an F 1-good filtration of ideals in R. If F 1 is m-primary we obtain sufficient conditions in order that the associated graded ring G(?) be Cohen–Macaulay. In the case where R is Gorenstein, we use the Cohen–Macaulay result to establish necessary and sufficient conditions for G(?) to be Gorenstein. We apply this result to the integral closure filtration ? associated to a monomial parameter ideal of a polynomial ring to give necessary and sufficient conditions for G(?) to be Gorenstein. Let (R, m) be a Gorenstein local ring, and let F 1 be an ideal with ht(F 1) = g > 0. If there exists a reduction J of ? with μ(J) = g and reduction number u: = r J (?), we prove that the extended Rees algebra R′(?) is quasi-Gorenstein with a-invariant b if and only if J n : F u = F n+b?u+g?1 for every n ∈ ?. Furthermore, if G(?) is Cohen–Macaulay, then the maximal degree of a homogeneous minimal generator of the canonical module ω G(?) is at most g and that of the canonical module ω R′(?) is at most g ? 1; moreover, R′(?) is Gorenstein if and only if J u : F u = F u . We illustrate with various examples cases where G(?) is or is not Gorenstein. 相似文献
10.
A weak Cayley table isomorphism is a bijection φ: G → H of groups such that φ(xy) ~ φ(x)φ(y) for all x, y ∈ G. Here ~denotes conjugacy. When G = H the set of all weak Cayley table isomorphisms φ: G → G forms a group 𝒲(G) that contains the automorphism group Aut(G) and the inverse map I: G → G, x → x ?1. Let 𝒲0(G) = ?Aut(G), I? ≤ 𝒲(G) and say that G has trivial weak Cayley table group if 𝒲(G) = 𝒲0(G). We show that all finite irreducible Coxeter groups (except possibly E 8) have trivial weak Cayley table group, as well as most alternating groups. We also consider some sporadic simple groups. 相似文献
11.
Let G be a finite simple graph on a vertex set V(G) = {x 11,…, x n1}. Also let m 1,…, m n ≥ 2 be integers and G 1,…, G n be connected simple graphs on the vertex sets V(G i ) = {x i1,…, x im i }. In this article, we provide necessary and sufficient conditions on G 1,…, G n for which the graph obtained by attaching the G i to G is unmixed or vertex decomposable. Then we characterize Cohen–Macaulay and sequentially Cohen–Macaulay graphs obtained by attaching the cycle graphs or connected chordal graphs to arbitrary graphs. 相似文献
12.
Basim Samir 《代数通讯》2013,41(6):2425-2436
Let α be an ordinal and κ be a cardinal, both infinite, such that κ ≤ |α|. For τ ∈αα, let sup(τ) = {i ∈ α: τ(i) ≠ i}. Let G κ = {τ ∈αα: |sup(τ)| < κ}. We consider variants of polyadic equality algebras by taking cylindrifications on Γ ? α, |Γ| < κ and substitutions restricted to G κ. Such algebras are also enriched with generalized diagonal elements. We show that for any variety V containing the class of representable algebas and satisfying a finite schema of equations, V fails to have the amalgamation property. In particular, many varieties of Halmos’ quasi-polyadic equality algebras and Lucas’ extended cylindric algebras (including that of the representable algebras) fail to have the amalgamation property. 相似文献
13.
We show that if A is a representation-finite selfinjective Artin algebra, then every P ? ? K b(𝒫 A ) with Hom K(Mod?A)(P ?,P ?[i]) = 0 for i ≠ 0 and add(P ?) = add(νP ?) is a direct summand of a tilting complex, and that if A, B are derived equivalent representation-finite selfinjective Artin algebras, then there exists a sequence of selfinjective Artin algebras A = B 0, B 1,…, B m = B such that, for any 0 ≤ i < m, B i+1 is the endomorphism algebra of a tilting complex for B i of length ≤ 1. 相似文献
14.
Davide Fusi 《代数通讯》2013,41(8):2989-3008
Let X be a smooth complex projective variety and let Z ? X be a smooth submanifold of dimension ≥ 2, which is the zero locus of a section of an ample vector bundle ? of rank dim X ? dim Z ≥ 2 on X. Let H be an ample line bundle on X, whose restriction H Z to Z is generated by global sections. The structure of triplets (X,?,H) as above is described under the assumption that the curve genus of the corank-1 vector bundle ? ⊕ H ⊕ (dim Z?1) is ≤ h 1( X ) + 2. 相似文献
15.
Let R be a semiprime ring with center Z(R), extended centroid C, U the maximal right ring of quotients of R, and m a positive integer. Let f: R → U be an additive m-power commuting map. Suppose that f is Z(R)-linear. It is proved that there exists an idempotent e ∈ C such that ef(x) = λx + μ(x) for all x ∈ R, where λ ∈C and μ: R → C. Moreover, (1 ? e)U ? M2(E), where E is a complete Boolean ring. As consequences of the theorem, it is proved that every additive, 2-power commuting map or centralizing map from R to U is commuting. 相似文献
16.
Let G be a non-abelian group and Z(G) be the center of G. The non-commuting graph Γ G associated to G is the graph whose vertex set is G?Z(G) and two distinct elements x, y are adjacent if and only if xy ≠ yx. We prove that if G and H are non-abelian nilpotent groups with irregular isomorphic non-commuting graphs, then |G| = |H|. 相似文献
17.
18.
Michael J. J. Barry 《代数通讯》2013,41(10):4231-4246
A Jordan partition λ(m, n, p) = (λ1, λ2, … , λ m ) is a partition of mn associated with the expression of a tensor V m ? V n of indecomposable KG-modules into a sum of indecomposables, where K is a field of characteristic p and G a cyclic group of p-power order. It is standard if λ i = m + n ? 2i + 1 for all i. We answer a recent question of Glasby, Praeger, and Xia who asked for necessary and sufficient conditions for λ(m, n, p) to be standard. 相似文献
19.
M. Ebrahimpour 《代数通讯》2013,41(9):3861-3875
Let R be a commutative ring with identity. We say that a proper ideal P of R is (n ? 1, n)-weakly prime (n ≥ 2) if 0 ≠ a 1…a n ∈ P implies a 1…a i?1 a i+1…a n ∈ P for some i ∈ {1,…, n}, where a 1,…, a n ∈ R. In this article, we study (n ? 1, n)-weakly prime ideals. A number of results concerning (n ? 1, n)-weakly prime ideals and examples of (n ? 1, n)-weakly prime ideals are given. Rings with the property that for a positive integer n such that 2 ≤ n ≤ 5, every proper ideal is (n ? 1, n)-weakly prime are characterized. Moreover, it is shown that in some rings, nonzero (n ? 1, n)-weakly prime ideals and (n ? 1, n)-prime ideals coincide. 相似文献
20.
Let G be a graph with vertex set V(G) and edge set E(G). Let k1, k2,…,km be positive integers. It is proved in this study that every [0,k1+…+km?m+1]‐graph G has a [0, ki]1m‐factorization orthogonal to any given subgraph H with m edges. © 2002 Wiley Periodicals, Inc. J Graph Theory 40: 267–276, 2002 相似文献