Supersolvable Frobenius groups with nilpotent centralizers

Let FH be a supersolvable Frobenius group with kernel F and complement H. Suppose that a finite group G admits FH as a group of automorphisms in such a manner that $C_G(F)=1$ and $C_G(H)$ is nilpotent of class c. We show that G is nilpotent of $(c,\left|FH\right|)$-bounded class.     

Vertex-primitive s-arc-transitive digraphs of linear groups

We study G-vertex-primitive and $(G,s)$-arc-transitive digraphs for almost simple groups G with socle ${\mathrm{PSL}}_n(q)$. We prove that $s?2$ for such digraphs, which provides the first step in determining an upper bound on s for all the vertex-primitive s-arc-transitive digraphs.     

Subgroup regular sets in Cayley graphs

Let Γ be a graph with vertex set V, and let a and b be nonnegative integers. A subset C of V is called an $(a,b)$-regular set in Γ if every vertex in C has exactly a neighbors in C and every vertex in $V?C$ has exactly b neighbors in C. In particular, $(0,1)$-regular sets and $(1,1)$-regular sets in Γ are called perfect codes and total perfect codes in Γ, respectively. A subset C of a group G is said to be an $(a,b)$-regular set of G if there exists a Cayley graph of G which admits C as an $(a,b)$-regular set. In this paper we prove that, for any generalized dihedral group G or any group G of order 4p or pq for some primes p and q, if a nontrivial subgroup H of G is a $(0,1)$-regular set of G, then it must also be an $(a,b)$-regular set of G for any $0?a?|H|?1$ and $0?b?|H|$ such that a is even when $|H|$ is odd. A similar result involving $(1,1)$-regular sets of such groups is also obtained in the paper.     

Hamiltonian and long paths in bipartite graphs with connectivity

Let G be a graph, $\nu (G)$ the order of G, $\kappa (G)$ the connectivity of G and k a positive integer such that $k\le (\nu (G)?2)/2$. Then G is said to be k-extendable if it has a matching of size k and every matching of size k extends to a perfect matching of G. A Hamiltonian path of a graph G is a spanning path of G. A bipartite graph G with vertex sets $V_1$ and $V_2$ is defined to be Hamiltonian-laceable if such that $|V_1|=|V_2|$ and for every pair of vertices $p\in V_1$ and $q\in V_2$, there exists a Hamiltonian path in G with endpoints p and q, or $|V_1|=|V_2|+1$ and for every pair of vertices $p,q\in V_1,p\ne q$, there exists a Hamiltonian path in G with endpoints p and q. Let G be a bipartite graph with bipartition $(X,Y)$. Define $bn(G)$ to be a maximum integer such that $0\le bn(G)<min\{|X|,|Y|\}$ and (1) for each non-empty subset S of X, if $|S|\le |X|?bn(G)$, then $|N(S)|\ge |S|+bn(G)$, and if $|X|?bn(G)<|S|\le |X|$, then $N(S)=Y$; and (2) for each non-empty subset S of Y, if $|S|\le |Y|?bn(G)$, then $|N(S)|\ge |S|+bn(G)$, and if $|Y|?bn(G)<|S|\le |Y|$, then $N(S)=X$; and (3) $bn(G)=0$ if there is no non-negative integer satisfying (1) and (2).Let G be a bipartite graph with bipartition $(X,Y)$ such that $|X|=|Y|$ and $bn(G)>0$. In this paper, we show that if $\nu (G)\le 2\kappa (G)+4bn(G)?4$, then G is Hamiltonian-laceable; or if $\nu (G)>6bn(G)?2$, then for every pair of vertices $x\in X$ and $y\in Y$, there is an $(x,y)$-path P in G with $|V(P)|\ge 6bn(G)?2$. We show some of its corollaries in k-extendable, bipartite graphs and a conjecture in k-extendable graphs.     

On base sizes for primitive groups of product type

Let $G?\mathrm{Sym}(\mathrm{\Omega })$ be a finite permutation group and recall that the base size of G is the minimal size of a subset of Ω with trivial pointwise stabiliser. There is an extensive literature on base sizes for primitive groups, but there are very few results for primitive groups of product type. In this paper, we initiate a systematic study of bases in this setting. Our first main result determines the base size of every product type primitive group of the form $L?P?\mathrm{Sym}(\mathrm{\Omega })$ with soluble point stabilisers, where $\mathrm{\Omega }={\mathrm{\Gamma }}^k$, $L?\mathrm{Sym}(\mathrm{\Gamma })$ and $P?S_k$ is transitive. This extends recent work of Burness on almost simple primitive groups. We also obtain an expression for the number of regular suborbits of any product type group of the form $L?P$ and we classify the groups with a unique regular suborbit under the assumption that P is primitive, which involves extending earlier results due to Seress and Dolfi. We present applications on the Saxl graphs of base-two product type groups and we conclude by establishing several new results on base sizes for general product type primitive groups.     

Universal deformation rings and self-injective Nakayama algebras

Let k be a field and let Λ be an indecomposable finite dimensional k-algebra such that there is a stable equivalence of Morita type between Λ and a self-injective split basic Nakayama algebra over k. We show that every indecomposable finitely generated Λ-module V has a universal deformation ring $R(\mathrm{\Lambda },V)$ and we describe $R(\mathrm{\Lambda },V)$ explicitly as a quotient ring of a power series ring over k in finitely many variables. This result applies in particular to Brauer tree algebras, and hence to p-modular blocks of finite groups with cyclic defect groups.     

On the weak 2-coloring number of planar graphs

For a graph $G=(V,E)$, a total ordering L on V, and a vertex $v\in V$, let ${\mathrm{Wcol}}_2[G,L,v]$ be the set of vertices $w\in V$ for which there is a path from v to w whose length is 0, 1 or 2 and whose L-least vertex is w. The weak 2-coloring number ${\mathrm{wcol}}_2(G)$ of G is the least k such that there is a total ordering L on V with $|{\mathrm{Wcol}}_2[G,L,v]|\le k$ for all vertices $v\in V$. We improve the known upper bound on the weak 2-coloring number of planar graphs from 28 to 23. As the weak 2-coloring number is the best known upper bound on the star list chromatic number of planar graphs, this bound is also improved.     

A note on the Hambleton-Taylor-Williams conjecture

I. Hambleton, L. Taylor and B. Williams conjectured a general formula in the spirit of H. Lenstra for the decomposition of $G_n(RG)$ for any finite group G and noetherian ring R. The conjectured decomposition was shown to hold for some large classes of finite groups. D. Webb and D. Yao discovered that the conjecture failed for the symmetric group $S_5$, but remarked that it still might be reasonable to expect the HTW-decomposition for solvable groups. In this paper we show that the solvable group $\mathrm{SL}(2,{\mathbb{F}}_3)$ is also a counterexample to the conjectured HTW-decomposition. Nevertheless, we prove that for any finite group G the rank of $G_1(\mathbb{Z}G)$ does not exceed the rank of the expression in the HTW-decomposition. We also show that the HTW-decomposition predicts correct torsion for $G_1(\mathbb{Z}G)$ for any finite group G. Furthermore, we prove that for any degree other than $n=1$ the conjecture gives a correct prediction for the rank of $G_n(\mathbb{Z}G)$.     

Bounds on multiplicities of spherical spaces over finite fields

Let G be a reductive group scheme of type A acting on a spherical scheme X. We prove that there exists a number C such that the multiplicity $\dim \mathrm{Hom}(\rho ,\mathbb{C}[X(F)])$ is bounded by C, for any finite field F and any irreducible representation ρ of $G(F)$. We give an explicit bound for C. We conjecture that this result is true for any reductive group scheme and when F ranges (in addition) over all local fields of characteristic 0.Different aspects of this conjecture were studied in [3], [11], [6], [7].