Group algebras

Given a group G and a commutative ring k with identity, one can define an k-algebra k[G] called the group algebra of G over k. An element α ∈ k[G] is said to be algebraic if f(α) = 0 for some non-zero polynomial f(X) ∈ k[X]. We will discuss some of the developments in the study of algebraic elements in group algebras.     

Commuting Graphs of Group Algebras

The commuting graph of a ring R, denoted by Γ(R), is a graph of all whose vertices are noncentral elements of R, and 2 distinct vertices x and y are adjacent if and only if xy = yx. In this article we investigate some graph-theoretic properties of Γ(kG), where G is a finite group, k is a field, and 0 ≠ |G| ∈k. Among other results it is shown that if G is a finite nonabelian group and k is an algebraically closed field, then Γ(kG) is not connected if and only if |G| = 6 or 8. For an arbitrary field k, we prove that Γ(kG) is connected if G is a nonabelian finite simple group or G′ ≠ G″ and G″ ≠ 1.     

The Projective Cover of the Trivial Module Over a Group Algebra of a Finite Group

We determine all finite groups G such that the Loewy length (socle length) of the projective cover P(k _G ) of the trivial kG-module k _G is four, where k is a field of characteristic p > 0 and kG is the group algebra of G over k, by using previous results and also the classification of finite simple groups. As a by-product we prove also that if p = 2 then all finite groups G such that the Loewy lengths of the principal block algebras of kG are four, are determined.     

Abelian Group Gradings on Full Matrix Rings

For positive integers ? and n, several authors studied ?_?-gradings of the full matrix ring M _n (k) over a field k. In this article, we show that every (G × H)-grading of M _n (k) can be constructed by a pair of compatible G-grading and H-grading of M _n (k), where G and H are any finite groups. When G and H are finite cyclic groups, we characterize all (G × H)-gradings which are isomorphic to a good grading. Moreover, the results can be generalized for any finite abelian group grading of M _n (k).     

<Emphasis Type="Italic">k</Emphasis>–Pyramidal One–Factorizations

We consider one–factorizations of complete graphs which possess an automorphism group fixing k ≥ 0 vertices and acting regularly (i.e., sharply transitively) on the others. Since the cases k = 0 and k = 1 are well known in literature, we study the case k≥ 2 in some detail. We prove that both k and the order of the group are even and the group necessarily contains k − 1 involutions. Constructions for some classes of groups are given. In particular we extend the result of [7]: let G be an abelian group of even order and with k − 1 involutions, a one–factorization of a complete graph admitting G as an automorphism group fixing k vertices and acting regularly on the others can be constructed.     

Group Gradings on Polynomial Algebras

Let k be a field. We consider gradings on a polynomial algebra k[X₁,…, X_n] by an arbitrary abelian group G, such that the indeterminates are homogeneous elements of nontrivial degree. We classify the isomorphism types of such gradings, and we count them in the case where G is finite. We present some examples of good gradings and find a minimal set of generators of the subalgebra of elements of trivial degree.     

Spanning k-Trees of n-Connected Graphs

A tree is called a k-tree if the maximum degree is at most k. We prove the following theorem, by which a closure concept for spanning k-trees of n-connected graphs can be defined. Let k ≥ 2 and n ≥ 1 be integers, and let u and v be a pair of nonadjacent vertices of an n-connected graph G such that deg _G (u) + deg _G (v) ≥ |G| − 1 − (k − 2)n, where |G| denotes the order of G. Then G has a spanning k-tree if and only if G + uv has a spanning k-tree.     

Group Actions,k-Derivations,and Finite Morphisms

Let G be an affine algebraic group over an algebraically closed field k of characteristic zero. In this article, we consider finite G-equivariant morphisms F:X → Y of irreducible affine G-varieties. First we determine under which conditions on Y the induced map F ^G :X//G → Y//G of quotient varieties is also finite. This result is reformulated in terms of kernels of derivations on k-algebras A ? B such that B is integral over A. Second we construct explicitly two examples of finite G-equivariant maps F. In the first one, F ^G is quasifinite but not finite. In the second one, F ^G is not even quasifinite.     

Semilocal Group Algebras

Let k[G] be a semilocal group algebra. It is shown that if k is an algebraically closed field, then every finitely generated flat k[G]-module is projective.     

Group connectivity of complementary graphs

Let G be a 2‐edge‐connected undirected graph, A be an (additive) abelian group and A* = A?{0}. A graph G is A‐connected if G has an orientation D(G) such that for every function b: V(G)?A satisfying , there is a function f: E(G)?A* such that for each vertex v∈V(G), the total amount of f values on the edges directed out from v minus the total amount of f values on the edges directed into v equals b(v). For a 2‐edge‐connected graph G, define Λ_g(G) = min{k: for any abelian group A with |A|?k, G is A‐connected }. In this article, we prove the following Ramsey type results on group connectivity:

1. Let G be a simple graph on n?6 vertices. If min{δ(G), δ(G^c)}?2, then either Λ_g(G)?4, or Λ_g(G^c)?4.
2. Let Z₃ denote the cyclic group of order 3, and G be a simple graph on n?44 vertices. If min{δ(G), δ(G^c)}?4, then either G is Z₃‐connected, or G^c is Z₃‐connected. © 2011 Wiley Periodicals, Inc. J Graph Theory

     

Block Primitive 2- (v, k, 1) Designs Admitting a Ree Simple Group

In this paper, we prove that if D is a 2- (v, k, 1) design withG  ≤  Aut(D) block primitive and soc (G)  =  ²G₂(q) then D is a Ree unital with parameters 2- (q³ +  1, q +  1, 1).     

A 2-factor with Short Cycles Passing Through Specified Independent Vertices in Graph

For a graph G, we define σ₂(G) := min{d(u) + d(v)|u, v ≠ ∈ E(G), u ≠ v}. Let k ≥ 1 be an integer and G be a graph of order n ≥ 3k. We prove if σ₂(G) ≥ n + k − 1, then for any set of k independent vertices v ₁,...,v _k , G has k vertex-disjoint cycles C ₁,..., C _k of length at most four such that v _i ∈ V(C _i ) for all 1 ≤ i ≤ k. And show if σ₂(G) ≥ n + k − 1, then for any set of k independent vertices v ₁,...,v _k , G has k vertex-disjoint cycles C ₁,..., C _k such that v _i ∈ V(C _i ) for all 1 ≤ i ≤ k, V(C ₁) ∪...∪ V(C _k ) = V(G), and |C _i | ≤ 4 for all 1 ≤ i ≤ k − 1. The condition of degree sum σ₂(G) ≥ n + k − 1 is sharp. Received: December 20, 2006. Final version received: December 12, 2007.     

On the admissibility of finite groups over global fields of finite characteristic

Let k be a global field of characteristic p. A finite group G is called k-admissible if there exists a division algebra finite dimensional and central over k which is a crossed product for G. Let G be a finite group with normal Sylow p-subgroup P. If the factor group G/P is k-admissible, then G is k-admissible. A necessary condition is given for a group to be k-admissible: if a finite group G is k-admissible, then every Sylow l-subgroup of G for l ≠ p is metacyclic with some additional restriction. Then it is proved that a metacyclic group G generated by x and y is k-admissible if some relation between x and y is satisfied.     

On the Perfect Matchings of Near Regular Graphs

Let k, h be positive integers with k ≤ h. A graph G is called a [k, h]-graph if k ≤ d(v) ≤ h for any v ? V(G){v \in V(G)}. Let G be a [k, h]-graph of order 2n such that k ≥ n. Hilton (J. Graph Theory 9:193–196, 1985) proved that G contains at least ?k/3?{\lfloor k/3\rfloor} disjoint perfect matchings if h = k. Hilton’s result had been improved by Zhang and Zhu (J. Combin. Theory, Series B, 56:74–89, 1992), they proved that G contains at least ?k/2?{\lfloor k/2\rfloor} disjoint perfect matchings if k = h. In this paper, we improve Hilton’s result from another direction, we prove that Hilton’s result is true for [k, k + 1]-graphs. Specifically, we prove that G contains at least ?\fracn3?+1+(k-n){\lfloor\frac{n}3\rfloor+1+(k-n)} disjoint perfect matchings if h = k + 1.     

Group connectivity in line graphs

Tutte introduced the theory of nowhere zero flows and showed that a plane graph G has a face k-coloring if and only if G has a nowhere zero A-flow, for any Abelian group A with |A|≥k. In 1992, Jaeger et al. [9] extended nowhere zero flows to group connectivity of graphs: given an orientation D of a graph G, if for any b:V(G)?A with ∑_v∈V(G)b(v)=0, there always exists a map f:E(G)?A−{0}, such that at each v∈V(G), in A, then G is A-connected. Let Z₃ denote the cyclic group of order 3. In [9], Jaeger et al. (1992) conjectured that every 5-edge-connected graph is Z₃-connected. In this paper, we proved the following.

• (i) 

  Every 5-edge-connected graph is Z3-connected if and only if every 5-edge-connected line graph is Z3-connected.

• (ii) 

  Every 6-edge-connected triangular line graph is Z3-connected.

• (iii) 

  Every 7-edge-connected triangular claw-free graph is Z3-connected.

In particular, every 6-edge-connected triangular line graph and every 7-edge-connected triangular claw-free graph have a nowhere zero 3-flow.     

Homology of balanced complexes via the Fourier transform

Let G ₀,…,G _k be finite abelian groups, and let G ₀∗⋯∗G _k be the join of the 0-dimensional complexes G _i . We give a characterization of the integral k-coboundaries of subcomplexes of G ₀∗⋯∗G _k in terms of the Fourier transform on the group G ₀×⋯×G _k . This provides a short proof of an extension of a recent result of Musiker and Reiner on a topological interpretation of the cyclotomic polynomial.     

Generic Idempotent Modules for a Finite Group

Let G be a finite group and k an algebraically closed field of characteristic p. Let F _U be the Rickard idempotent k G-module corresponding to the set U of subvarieties of the cohomology variety V _G which are not irreducible components. We show that F _U is a finite sum of generic modules corresponding to the irreducible components of V _G . In this context, a generic module is an indecomposable module of infinite length over k G but finite length as a module over its endomorphism ring.     

Group Weighted Matchings in Bipartite Graphs

Let G be a bipartite graph with bicoloration {A, B}, |A| = |B|, and let w : E(G) K where K is a finite abelian group with k elements. For a subset S E(G) let . A Perfect matching M E(G) is a w-matching if w(M) = 1.A characterization is given for all w's for which every perfect matching is a w-matching.It is shown that if G = K _k+1,k+1 then either G has no w-matchings or it has at least 2 w-matchings.If K is the group of order 2 and deg(a) d for all a A, then either G has no w-matchings, or G has at least (d – 1)! w-matchings.R. Meshulam: Research supported by a Technion V.P.R. Grant No. 100-854.