Which finite simple groups are unit groups?

We prove that if G$G$ is a finite simple group which is the unit group of a ring, then G$G$ is isomorphic to: (a) a cyclic group of order 2; or (b) a cyclic group of prime order 2^k−1$2^k-1$ for some k$k$; or (c) a projective special linear group _PSLn(_F2)${PSL}_n\left({\mathbb{F}}_2\right)$ for some n≥3$n\ge 3$. Moreover, these groups do all occur as unit groups. We deduce this classification from a more general result, which holds for groups G$G$ with no non-trivial normal 2-subgroup.     

Generalized cover ideals and the persistence property

Let I   be a square-free monomial ideal in R=k[_x1,…,_xn]$R=k[x_1,\dots ,x_n]$, and consider the sets of associated primes Ass(I^s)$\mathrm{Ass}(I^s)$ for all integers s?1$s?1$. Although it is known that the sets of associated primes of powers of I eventually stabilize, there are few results about the power at which this stabilization occurs (known as the index of stability). We introduce a family of square-free monomial ideals that can be associated to a finite simple graph G that generalizes the cover ideal construction. When G   is a tree, we explicitly determine Ass(I^s)$\mathrm{Ass}(I^s)$ for all s?1$s?1$. As consequences, not only can we compute the index of stability, we can also show that this family of ideals has the persistence property.     

Exceptional collections of line bundles on projective homogeneous varieties

We construct new examples of exceptional collections of line bundles on the variety of Borel subgroups of a split semisimple linear algebraic group G$G$ of rank 2 over a field. We exhibit exceptional collections of the expected length for types _A2$A_2$ and _B2=_C2$B_2=C_2$ and prove that no such collection exists for type _G2$G_2$. This settles the question of the existence of full exceptional collections of line bundles on projective homogeneous G$G$-varieties for split linear algebraic groups G$G$ of rank at most 2.     

First order theory of cyclically ordered groups

By a result known as Rieger's theorem (1956), there is a one-to-one correspondence, assigning to each cyclically ordered group H a pair $(G,z)$ where G is a totally ordered group and z is an element in the center of G, generating a cofinal subgroup $〈z〉$ of G, and such that the cyclically ordered quotient group $G/〈z〉$ is isomorphic to H. We first establish that, in this correspondence, the first-order theory of the cyclically ordered group H is uniquely determined by the first-order theory of the pair $(G,z)$. Then we prove that the class of cyclically orderable groups is an elementary class and give an axiom system for it. Finally we show that, in contrast to the fact that all theories of totally ordered Abelian groups have the same universal part, there are uncountably many universal theories of Abelian cyclically ordered groups.     

Normal subgroups of profinite groups of non-negative deficiency

The principal goal of the paper is to show that the existence of a finitely generated normal subgroup of infinite index in a profinite group G of non-negative deficiency gives rather strong consequences for the structure of G. To make this precise we introduce the notion of p-deficiency (p a prime) for a profinite group G. We prove that if the p-deficiency of G is positive and N is a finitely generated normal subgroup such that the p  -Sylow subgroup of G/N$G/N$ is infinite and p divides the order of N   then we have _cdp(G)=2${\mathrm{cd}}_p(G)=2$, _cdp(N)=1${\mathrm{cd}}_p(N)=1$ and _vcdp(G/N)=1${\mathrm{vcd}}_p(G/N)=1$ for the cohomological p-dimensions; moreover either the p  -Sylow subgroup of G/N$G/N$ is virtually cyclic or the p-Sylow subgroup of N is cyclic. If G is a profinite Poincaré duality group of dimension 3 at a prime p   (PD³${\mathit{PD}}^3$-group at p) we show that for N and p as above either N   is PD¹${\mathit{PD}}^1$ at p   and G/N$G/N$ is virtually PD²${\mathit{PD}}^2$ at p or N   is PD²${\mathit{PD}}^2$ at p   and G/N$G/N$ is virtually PD¹${\mathit{PD}}^1$ at p.     

The Zassenhaus filtration,Massey products,and representations of profinite groups

We consider the p  -Zassenhaus filtration (_Gn)$(G_n)$ of a profinite group G  . Suppose that G=S/N$G=S/N$ for a free profinite group S and a normal subgroup N of S   contained in _Sn$S_n$. Under a cohomological assumption on the n-fold Massey products (which holds, e.g., if G has p  -cohomological dimension ≤ 1), we prove that _Gn+1$G_{n+1}$ is the intersection of all kernels of upper-triangular unipotent (n+1)$(n+1)$-dimensional representations of G   over _Fp${\mathbb{F}}_p$. This extends earlier results by Miná?, Spira, and the author on the structure of absolute Galois groups of fields.     

On words that are concise in residually finite groups

A group-word w$w$ is called concise if whenever the set of w$w$-values in a group G$G$ is finite it always follows that the verbal subgroup w(G)$w\left(G\right)$ is finite. More generally, a word w$w$ is said to be concise in a class of groups X$X$ if whenever the set of w$w$-values is finite for a group G∈X$G\in X$, it always follows that w(G)$w\left(G\right)$ is finite. P. Hall asked whether every word is concise. Due to Ivanov the answer to this problem is known to be negative. Dan Segal asked whether every word is concise in the class of residually finite groups. In this direction we prove that if w$w$ is a multilinear commutator and q$q$ is a prime-power, then the word w^q$w^q$ is indeed concise in the class of residually finite groups. Further, we show that in the case where w=_γk$w={\gamma }_k$ the word w^q$w^q$ is boundedly concise in the class of residually finite groups. It remains unknown whether the word w^q$w^q$ is actually concise in the class of all groups.     

On oriented graphs whose skew spectral radii do not exceed 2

Let S(G^σ)$S(G^{\sigma })$ be the skew adjacency matrix of the oriented graph G^σ$G^{\sigma }$ of order n   and _λ1,_λ2,…,_λn${\lambda }_1,{\lambda }_2,\dots ,{\lambda }_n$ be all eigenvalues of S(G^σ)$S(G^{\sigma })$. The skew spectral radius _ρs(G^σ)${\rho }_s(G^{\sigma })$ of G^σ$G^{\sigma }$ is defined as max{|_λ1|,|_λ2|,…,|_λn|}$\max \{|{\lambda }_1|,|{\lambda }_2|,\dots ,|{\lambda }_n|\}$. In this paper, we investigate oriented graphs whose skew spectral radii do not exceed 2.