Exponent of a finite group with a four-group of automorphisms

Let A be a group isomorphic with either S ₄, the symmetric group on four symbols, or D ₈, the dihedral group of order 8. Let V be a normal four-subgroup of A and ?? an involution in ${A\setminus V}$ . Suppose that A acts on a finite group G in such a manner that C _G (V)?=?1 and C _G (??) has exponent e. We show that if ${A\cong S_4}$ then the exponent of G is e-bounded and if ${A\cong D_8}$ then the exponent of the derived group G?? is e-bounded. This work was motivated by recent results on the exponent of a finite group admitting an action by a Frobenius group of automorphisms.     

Structures and Chromaticity of Extremal 3-Colourable Sparse Graphs

 Assume that G is a 3-colourable connected graph with e(G) = 2v(G) −k, where k≥ 4. It has been shown that s ₃(G) ≥ 2 ^{k −3}, where s _r (G) = P(G,r)/r! for any positive integer r and P(G, λ) is the chromatic polynomial of G. In this paper, we prove that if G is 2-connected and s ₃(G) < 2 ^{k −2}, then G contains at most v(G) −k triangles; and the upper bound is attained only if G is a graph obtained by replacing each edge in the k-cycle C _k by a 2-tree. By using this result, we settle the problem of determining if W(n, s) is χ-unique, where W(n, s) is the graph obtained from the wheel W _n by deleting all but s consecutive spokes. Received: January 29, 1999 Final version received: April 8, 2000     

Finite index subgroups in profinite groups

We prove that every subgroup of finite index in a (topologically) finitely generated profinite group is open. This implies that the topology in such a group is uniquely determined by the group structure. The result follows from a ‘uniformity theorem’ about finite groups: given a group word w that defines a locally finite variety and a natural number d, there exists f=f_w(d) such that in every finite d-generator group G, each element of the verbal subgroup w(G) is a product of fw-values. Similar methods show that in a finite d-generator group, each element of the derived group is a product of g(d) commutators; this implies that the (abstract) derived group in any finitely generated profinite group is closed. To cite this article: N. Nikolov, D. Segal, C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

[r,s,t]-Coloring of K n,n

Let G=(V(G),E(G)) be a simple graph. Given non-negative integers r,s, and t, an [r,s,t]-coloring of G is a mapping c from V(G)∪E(G) to the color set {0,1,…,k?1} such that |c(v _i )?c(v _j )|≥r for every two adjacent vertices v _i ,v _j , |c(e _i )?c(e _j )|≥s for every two adjacent edges e _i ,e _j , and |c(v _i )?c(e _j )|≥t for all pairs of incident vertices and edges, respectively. The [r,s,t]-chromatic number χ _r,s,t (G) of G is defined to be the minimum k such that G admits an [r,s,t]-coloring. We determine χ _r,s,t (K _n,n ) in all cases.     

Procyclic coverings of commutators in profinite groups

We consider profinite groups in which all commutators are contained in a union of finitely many procyclic subgroups. It is shown that if G is a profinite group in which all commutators are covered by m procyclic subgroups, then G possesses a finite characteristic subgroup M contained in G′ such that the order of M is m-bounded and G′/M is procyclic. If G is a pro-p group such that all commutators in G are covered by m procyclic subgroups, then G′ is either finite of m-bounded order or procyclic.     

Centralizers of coprime automorphisms of finite groups

Let A be an elementary abelian group of order p ^k with k ≥ 3 acting on a finite p′-group G. The following results are proved. If γ _k-2(C _G (a)) is nilpotent of class at most c for any ${a \in A^{\#}}$ , then γ _k-2(G) is nilpotent and has {c, k, p}-bounded nilpotency class. If, for some integer d such that 2 ^d  + 2 ≤ k, the dth derived group of C _G (a) is nilpotent of class at most c for any ${a \in A^{\#}}$ , then the dth derived group G ^(d) is nilpotent and has {c, k, p}-bounded nilpotency class.     

Exponent of a finite group with a fixed-point-free four-group of automorphisms

Let e be a positive integer and G a finite group acted on by the four-group V in such a manner that C _G (V) = 1. Suppose that V contains an element v such that the centralizer C _G (v) has exponent e. Then the exponent of G″, the second derived group of G, is bounded in terms of e only.     

Normal coverings of solvable groups

For a finite non cyclic group G, let γ(G) be the smallest integer k such that G contains k proper subgroups H ₁, . . . , H _k with the property that every element of G is contained in \({H_i^g}\) for some \({i \in \{1,\dots,k\}}\) and \({g \in G.}\) We prove that for every n ≥ 2, there exists a finite solvable group G with γ(G) = n.     

Quadratic function fields with invariant class group

Emil Artin studied quadratic extensions of k(x) where k is a prime field of odd characteristic. He showed that there are only finitely many such extensions in which the ideal class group has exponent two and the infinite prime does not decompose. The main result of this paper is: If K is a quadratic imaginary extension of k(x) of genus G, where k is a finite field of order q, in which the infinite prime of k(x) ramifies, and if the ideal class group has exponent 2, then q = 9, 7, 5, 4, 3, or 2 and G ≤ 1, 1, 2, 2, 4, and 8, respectively. The method of Artin's proof gives G ≤ 13, 9, and 9724 for q = 7, 5, and 3, respectively. If the infinite prime is inert in K, both the methods of this paper and Artin's methods give bounds on the genus that are roughly double those in the ramified case.     

Secure domination critical graphs

A secure dominating set X of a graph G is a dominating set with the property that each vertex u∈V_G−X is adjacent to a vertex v∈X such that (X−{v})∪{u} is dominating. The minimum cardinality of such a set is called the secure domination number, denoted by γ_s(G). We are interested in the effect of edge removal on γ_s(G), and characterize γ_s-ER-critical graphs, i.e. graphs for which γ_s(G−e)>γ_s(G) for any edge e of G, bipartite γ_s-ER-critical graphs and γ_s-ER-critical trees.     

Multiplicative infinite loop space theory

Let G be a finitely presented group given by its pre-abelian presentation <X1,…,Xm; Xe11ζ1,…,Xemmζ,ζm+1,…>, where ei≥0 for i = 1,…, m and ζj?G′ for j≥1. Let N be the subgroup of G generated by the normal subgroups [xeii, G] for i = 1,…, m. Then Dn+2(G)≡γn+2(G) (modNG′) for all n≥0, where G” is the second commutator subgroup of G,γn+2(G) is the (n+2)th term of the lower central series of G and Dn+2(G) = G∩(1+△n+2(G)) is the (n+2)th dimension subgroup of G.     

Bounds on some van der Waerden numbers

For positive integers s and k₁,k₂,…,ks, the van der Waerden number w(k₁,k₂,…,ks;s) is the minimum integer n such that for every s-coloring of set {1,2,…,n}, with colors 1,2,…,s, there is a ki-term arithmetic progression of color i for some i. We give an asymptotic lower bound for w(k,m;2) for fixed m. We include a table of values of w(k,3;2) that are very close to this lower bound for m=3. We also give a lower bound for w(k,k,…,k;s) that slightly improves previously-known bounds. Upper bounds for w(k,4;2) and w(4,4,…,4;s) are also provided.     

Sets in Abelian groups with distinct sums of pairs

A subset S={s₁,…,sk} of an Abelian group G is called an St-set of size k if all sums of t different elements in S are distinct. Let s(G) denote the cardinality of the largest S₂-set in G. Let v(k) denote the order of the smallest Abelian group for which s(G)?k. In this article, bounds for s(G) are developed and v(k) is determined for k?15 by computing s(G) for Abelian groups of order up to 183 using exhaustive backtrack search with isomorph rejection.     

On the set of discrete subgroups of bounded covolume in a semisimple group

In this noteG is a locally compact group which is the product of finitely many groups G_s(k_s)(s∈S), where k_s is a local field of characteristic zero and G_s an absolutely almost simplek _s-group, ofk _s-rank ≥1. We assume that the sum of the r_s is ≥2 and fix a Haar measure onG. Then, given a constantc > 0, it is shown that, up to conjugacy,G contains only finitely many irreducible discrete subgroupsL of covolume ≥c (4.2). This generalizes a theorem of H C Wang for real groups. His argument extends to the present case, once it is shown thatL is finitely presented (2.4) and locally rigid (3.2).     

Domination critical graphs

A set of vertices S is said to dominate the graph G if for each v ? S, there is a vertex u ∈ S with u adjacent to v. The smallest cardinality of any such dominating set is called the domination number of G and is denoted by γ(G). The purpose of this paper is to initiate an investigation of those graphs which are critical in the following sense: For each v, u ∈ V(G) with v not adjacent to u, γ(G + vu) < γ(G). Thus G is k-y-critical if γ(G) = k and for each edge e ? E(G), γ(G + e) = k ?1. The 2-domination critical graphs are characterized the properties of the k-critical graphs with k ≥ 3 are studied. In particular, the connected 3-critical graphs of even order are shown to have a 1-factor and some stringent restrictions on their degree sequences and diameters are obtained.     

A characterization of the linear groups L 2(p)

Let G be a finite group and π _e (G) be the set of element orders of G. Let k ∈ π _e (G) and m _k be the number of elements of order k in G. Set nse(G):= {m _k : k ∈ π _e (G)}. In fact nse(G) is the set of sizes of elements with the same order in G. In this paper, by nse(G) and order, we give a new characterization of finite projective special linear groups L ₂(p) over a field with p elements, where p is prime. We prove the following theorem: If G is a group such that |G| = |L ₂(p)| and nse(G) consists of 1, p ² ? 1, p(p + ?)/2 and some numbers divisible by 2p, where p is a prime greater than 3 with p ≡ 1 modulo 4, then G ? L ₂(p).     

γ-Bounded representations of amenable groups

Let G be an amenable group, let X be a Banach space and let π:G→B(X) be a bounded representation. We show that if the set is γ-bounded then π extends to a bounded homomorphism w:C^∗(G)→B(X) on the group C^∗-algebra of G. Moreover w is necessarily γ-bounded. This extends to the Banach space setting a theorem of Day and Dixmier saying that any bounded representation of an amenable group on Hilbert space is unitarizable. We obtain additional results and complements when G=Z, R or T, and/or when X has property (α).     

On a geometrical method of construction of maximal t-linearly independent sets

The problem of constructing a maximal t-linearly independent set in V(r; s) (called a maximal L_t(r, s)-set in this paper) is a very important one (called a packing problem) concerning a fractional factorial design and an error correcting code where V(r; s) is an r-dimensional vector space over a Galois field GF(s) and s is a prime or a prime power. But it is very difficult to solve it and attempts made by several research workers have been successful only in special cases.In this paper, we introduce the concept of a {Σ_α=1^kw_α, m; t, s}-min · hyper with weight (w₁, w₂,…, w_k) and using this concept and the structure of a finite projective geometry PG(n ? 1, s), we shall give a geometrical method of constructing a maximal L_t(t + r, s)-set with length t + r + n for any integers r, n, and s such that n ? 3, n ? 1 ? r₀ ? n + s ? 2 and r₁ ? 1, where r = (r₁ + 1)v_n?1 ? r₀ and $\text{v}{}_{\mathrm{n}}\text{=}\text{(s}{}^{\mathrm{n}}\text{? 1)}\text{(s ? 1)}$.     

The equivariant topology of stable Kneser graphs

The stable Kneser graph SG_n,k, n?1, k?0, introduced by Schrijver (1978) [19], is a vertex critical graph with chromatic number k+2, its vertices are certain subsets of a set of cardinality m=2n+k. Björner and de Longueville (2003) [5] have shown that its box complex is homotopy equivalent to a sphere, Hom(K₂,SG_n,k)?Sk. The dihedral group D2m acts canonically on SG_n,k, the group C₂ with 2 elements acts on K₂. We almost determine the (C₂×D2m)-homotopy type of Hom(K₂,SG_n,k) and use this to prove the following results.The graphs SG_2s,4 are homotopy test graphs, i.e. for every graph H and r?0 such that Hom(SG_2s,4,H) is (r−1)-connected, the chromatic number χ(H) is at least r+6.If k∉{0,1,2,4,8} and n?N(k) then SG_n,k is not a homotopy test graph, i.e. there are a graph G and an r?1 such that Hom(SG_n,k,G) is (r−1)-connected and χ(G)<r+k+2.     

On graphs critical with respect to vertex partition numbers

For k?0, ?_k(G) denotes the Lick-White vertex partition number of G. A graph G is called (n, k)-critical if it is connected and for each edge e of G?_k(G–e)<?_k(G)=n. We describe all (2, k)-critical graphs and for n?3,k?1 we extend and simplify a result of Bollobás and Harary giving one construction of a family of (n, k)-critical graphs of every possible order.