Minimal presentations for certain group extensions

IfG is a finite group thend(G) denotes the minimal number of generators ofG. IfH andK are groups then the extension, 1 →H →G →K → 1, is called an outer extension ofK byH ifd(G)=d(H)+d(K). Let be the class of groups containing all finitep-groupsG which has a presentation withd(G) = dimH ¹(G,z _p ) generators andr(G)=dimH ² (G,Z _p ) relations: in this article it is shown that ifK is a non cyclic group belonging to andH is a finite abelian p-group then any outer extension ofK byH belongs to .     

On <Emphasis Type="Italic">k</Emphasis>-pairable graphs from trees

The concept of the k-pairable graphs was introduced by Zhibo Chen (On k-pairable graphs, Discrete Mathematics 287 (2004), 11–15) as an extension of hypercubes and graphs with an antipodal isomorphism. In the same paper, Chen also introduced a new graph parameter p(G), called the pair length of a graph G, as the maximum k such that G is k-pairable and p(G) = 0 if G is not k-pairable for any positive integer k. In this paper, we answer the two open questions raised by Chen in the case that the graphs involved are restricted to be trees. That is, we characterize the trees G with p(G) = 1 and prove that p(G □ H) = p(G) + p(H) when both G and H are trees.     

On r-recognition by prime graph of Bp(3) where p is an odd prime

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 @ B_p(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.     

On automorphisms of some finite <Emphasis Type="Italic">p</Emphasis>-groups

We give a sufficient condition on a finite p-group G of nilpotency class 2 so that Aut _c (G) = Inn(G), where Aut _c (G) and Inn(G) denote the group of all class preserving automorphisms and inner automorphisms of G respectively. Next we prove that if G and H are two isoclinic finite groups (in the sense of P. Hall), then Aut _c (G) ≃ Aut _c (H). Finally we study class preserving automorphisms of groups of order p ⁵, p an odd prime and prove that Aut _c (G) = Inn(G) for all the groups G of order p ⁵ except two isoclinism families.     

Zero-sum problems with congruence conditions

For a finite abelian group G and a positive integer d, let s _dℕ(G) denote the smallest integer ℓ∈ℕ₀ such that every sequence S over G of length |S|≧ℓ has a nonempty zero-sum subsequence T of length |T|≡0 mod d. We determine s _dℕ(G) for all d≧1 when G has rank at most two and, under mild conditions on d, also obtain precise values in the case of p-groups. In the same spirit, we obtain new upper bounds for the Erdős–Ginzburg–Ziv constant provided that, for the p-subgroups G _p of G, the Davenport constant D(G _p ) is bounded above by 2exp  (G _p )−1. This generalizes former results for groups of rank two.     

Longest Paths and Longest Cycles in Graphs with Large Degree Sums

 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     

Relative length of long paths and cycles in graphs with large degree sums

For a graph G, p(G) denotes the order of a longest path in G and c(G) the order of a longest cycle. We show that if G is a connected graph n ≥ 3 vertices such that d(u) + d(v) + d(w) ≧ n for all triples u, v, w of independent vertices, then G satisfies c(G) ≥ p(G) – 1, or G is in one of six families of exceptional graphs. This generalizes results of Bondy and of Bauer, Morgana, Schmeichel, and Veldman. © 1995, John Wiley & Sons, Inc.     

Primitives and central detection numbers in group cohomology

Fix a prime p. Given a finite group G, let H^∗(G) denote its mod p cohomology. In the early 1990s, Henn, Lannes, and Schwartz introduced two invariants d₀(G) and d₁(G) of H^∗(G) viewed as a module over the mod p Steenrod algebra. They showed that, in a precise sense, H^∗(G) is respectively detected and determined by Hd(CG(V)) for d?d₀(G) and d?d₁(G), with V running through the elementary abelian p-subgroups of G.The main goal of this paper is to study how to calculate these invariants. We find that a critical role is played by the image of the restriction of H^∗(G) to H^∗(C), where C is the maximal central elementary abelian p-subgroup of G. A measure of this is the top degree e(G) of the finite dimensional Hopf algebra H^∗(C)_⊗H^∗(G)Fp, a number that tends to be quite easy to calculate.Our results are complete when G has a p-Sylow subgroup P in which every element of order p is central. Using the Benson-Carlson duality, we show that in this case, d₀(G)=d₀(P)=e(P), and a similar exact formula holds for d₁. As a bonus, we learn that He(G)(P) contains nontrivial essential cohomology, reproving and sharpening a theorem of Adem and Karagueuzian.In general, we are able to show that d₀(G)?max{e(CG(V))|V<G} if certain cases of Benson's Regularity Conjecture hold. In particular, this inequality holds for all groups such that the difference between the p-rank of G and the depth of H^∗(G) is at most 2. When we look at examples with p=2, we learn that d₀(G)?14 for all groups with 2-Sylow subgroup of order up to 64, with equality realized when G=SU(3,4).En route we study two objects of independent interest. If C is any central elementary abelian p-subgroup of G, then H^∗(G) is an H^∗(C)-comodule, and we prove that the subalgebra of H^∗(C)-primitives is always Noetherian of Krull dimension equal to the p-rank of G minus the p-rank of C. If the depth of H^∗(G) equals the rank of Z(G), we show that the depth essential cohomology of G is nonzero (reproving and extending a theorem of Green), and Cohen-Macauley in a certain sense, and prove related structural results.     

Weak factorizations of operators in the group von Neumann algebras of certain amenable groups and applications

Let G be a compact group whose local weight b(G) has uncountable cofinality. Let H be an amenable locally compact group and A(G × H) be the Fourier algebra of G × H. We prove that the group von Neumann algebra VN(G × H) = A(G × H)^* has the weak uniform A(G × H)^** factorization property of level b(G). As a corollary we show that A(G × H) is strongly Arens irregular, and the topological centre of UC ₂(G × H)^* is equal to the Fourier–Stieltjes algebra B(G × H).     

The Fractional Chromatic Number Of The Categorical Product Of Graphs

We prove that the identity

On the nonabelian tensor square and capability of groups of order <Emphasis Type="Italic">p</Emphasis><Superscript>2</Superscript><Emphasis Type="Italic">q</Emphasis>

A group G is said to be capable if it is isomorphic to the central factor group H/Z(H) for some group H. Let G be a nonabelian group of order p ² q for distinct primes p and q. In this paper, we compute the nonabelian tensor square of the group G. It is also shown that G is capable if and only if either Z(G) = 1 or p < q and G^ab=\mathbbZ_p×\mathbbZ_p{G^{\rm ab}=\mathbb{Z}_{p}\times\mathbb{Z}_{p}} .     

On the order of certain close to regular graphs without a matching of given size

A graph G is a {d, d+k}-graph, if one vertex has degree d+k and the remaining vertices of G have degree d. In the special case of k = 0, the graph G is d-regular. Let k, p ⩾ 0 and d, n ⩾ 1 be integers such that n and p are of the same parity. If G is a connected {d, d+k{-graph of order n without a matching M of size 2|M| = n − p, then we show in this paper the following: If d = 2, then k ⩾ 2(p + 2) and

A note on <Emphasis Type="Italic">p</Emphasis>-groups of order ≤ <Emphasis Type="Italic">p</Emphasis><Superscript>4</Superscript>

In [1], we defined c(G), q(G) and p(G). In this paper we will show that if G is a p-group, where p is an odd prime and |G| ≤ p ⁴, then c(G) = q(G) = p(G). However, the question of whether or not there is a p-group G with strict inequality c(G) = q(G) < p(G) is still open.     

On super edge-antimagic total labelings of

A graph G of order p and size q is called (a,d)-edge-antimagic total if there exists a bijective function f:V(G)E(G)→{1,2,…,p+q} such that the edge-weights w(uv)=f(u)+f(v)+f(uv), uvE(G), form an arithmetic sequence with first term a and common difference d. The graph G is said to be super (a,d)-edge-antimagic total if the vertex labels are 1,2,…,p. In this paper we study super (a,d)-edge-antimagic properties of mK_n, that is, of the graph formed by the disjoint union of m copies of K_n.     

Associated primes for cohomology modules

Let G be a finite group acting linearly on a finite dimensional vector space V defined over a field k of characteristic p, where p is assumed to divide the group order. Let R := S(V *) be the symmetric algebra of the dual on which G acts naturally by algebra automorphisms. We study the R^G-modules Hⁱ(G, R) for i > 0. In particular we give a formula which describes the annihilator of a general element of Hⁱ(G, R) in terms of the relative transfer ideals of R^G, and consequently prove that the associated primes of these cohomology modules are equal to the radicals of certain relative transfer ideals. Received: 5 June 2008     

On Optimal Orientations of Cartesian Products of Trees

. Let d(D) (resp., d(G)) denote the diameter and r(D) (resp., r(G)) the radius of a digraph D (resp., graph G). Let G×H denote the cartesian product of two graphs G and H. An orientation D of G is said to be (r, d)-invariant if r(D)=r(G) and d(D)=d(G). Let {T _i }, i=1,…,n, where n≥2, be a family of trees. In this paper, we show that the graph ∏ _{i =1} ⁿ T _i admits an (r, d)-invariant orientation provided that d(T ₁)≥d(T ₂)≥4 for n=2, and d(T ₁)≥5 and d(T ₂)≥4 for n≥3. Received: July 30, 1997 Final version received: April 20, 1998     

The emergence of a giant component in random subgraphs of pseudo‐random graphs

Let G be a d‐regular graph G on n vertices. Suppose that the adjacency matrix of G is such that the eigenvalue λ which is second largest in absolute value satisfies λ = o(d). Let G_p with p = α/d be obtained from G by including each edge of G independently with probability p. We show that if α < 1, then whp the maximum component size of G_p is O(log n) and if α > 1, then G_p contains a unique giant component of size Ω(n), with all other components of size O(log n). © 2003 Wiley Periodicals, Inc. Random Struct. Alg., 2004     

The p-Bondage Number of Trees

Let p be a positive integer and G = (V, E) be a simple graph. A p-dominating set of G is a subset D  í  V{D\,{\subseteq}\, V} such that every vertex not in D has at least p neighbors in D. The p-domination number of G is the minimum cardinality of a p-dominating set of G. The p-bondage number of a graph G with (ΔG) ≥ p is the minimum cardinality among all sets of edges B í E{B\subseteq E} for which γ _p (G − B) > γ _p (G). For any integer p ≥ 2 and tree T with (ΔT) ≥ p, this paper shows that 1 ≤  b _p (T) ≤ (ΔT) − p + 1, and characterizes all trees achieving the equalities.     

Bounded Direction–Length Frameworks

A direction–length framework is a pair (G,p) where G=(V;D,L) is a ‘mixed’ graph whose edges are labelled as ‘direction’ or ‘length’ edges and p is a map from V to ℝ ^d for some d. The label of an edge uv represents a direction or length constraint between p(u) and p(v). Let G ⁺ be obtained from G by adding, for each length edge e of G, a direction edge with the same end vertices as e. We show that (G,p) is bounded if and only if (G ⁺,p) is infinitesimally rigid. This gives a characterization of when (G,p) is bounded in terms of the rank of the rigidity matrix of (G ⁺,p). We use this to characterize when a mixed graph is generically bounded in ℝ ^d . As an application we deduce that if (G,p) is a globally rigid generic framework with at least two length edges and e is a length edge of G then (G∖e,p) is bounded.     

On a group with three supersolvable subgroups of pairwise relatively prime indices

In this paper, we show that if G is a finite group with three supersolvable subgroups of pairwise relatively prime indices in G and G′ is nilpotent, then G is supersolvable. Let π(G) denote the set of prime divisors of |G| and max(π(G)) denote the largest prime divisor of |G|. We also establish that if G is a finite group such that G has three supersolvable subgroups H, K, and L whose indices in G are pairwise relatively prime, q \nmid p-1{q \nmid p-1} where p =  max(π(G)) and q = max(π(L)) with L a Hall p′-subgroup of G, then G is supersolvable.