Minimal covers of Sn by abelian subgroups and maximal subsets of pairwise noncommuting elements, II

AbstractLet βn denote the minimum possible cardinality of a cover of the symmetric group Sn by abelian subgroups and let αn denote the maximum possible cardinality of a set of pairwise noncommuting elements of Sn. Then αn ≠ βn for all n ⩾ 15. This implies that the sequence (βnαn := 1, 2, …) takes on infinitely many distinct values and does not converge

On commuting and non-commuting complexes

In this paper we study various simplicial complexes associated to the commutative structure of a finite group G. We define NC(G) (resp. C(G)) as the complex associated to the poset of pairwise non-commuting (resp. commuting) sets in G. We observe that NC(G) has only one positive dimensional connected component, which we call BNC(G), and we prove that BNC(G) is simply connected. Our main result is a simplicial decomposition formula for BNC(G) which follows from a result of A. Bjorner, M. Wachs and V. Welker on inflated simplicial complexes. As a corollary, we obtain that if G has a nontrivial center or if G has odd order, then the homology group H_{n-1}(BNC(G)) is nontrivial for every n such that G has a maximal noncommuting set of order n

Coverings of finite groups by few proper subgroups

A connection between maximal sets of pairwise non-commuting elements and coverings of a finite group by proper subgroups is established. This allows us to study coverings of groups by few proper subgroups. The p-groups without p+2 pairwise non-commuting elements are classified. We also prove that if a p-group admits an irredundant covering by p+2 subgroups, then p=2. Some related topics are also discussed

Abelian covers of alternating groups

Let G = A n, a finite alternating group. We study the commuting graph of G and establish, for all possible values of n barring 13, 14, 17, and 19 whether or not the independence number is equal to the clique-covering number

Nilpotent covers and non-nilpotent subsets of finite groups of Lie type

Let~$G$ be a finite group, and~$c$ an element of~\longintegers. A subgroup~$H$ of~$G$ is said to be {\it $c$-nilpotent} if it is nilpotent, and has nilpotency class at most~$c$. A subset~$X$ of~$G$ is said to be {\it non-$c$-nilpotent} if it contains no two elements~$x$ and~$y$ such that the subgroup $\langle x,y\rangle$ is $c$-nilpotent. In this paper we study the quantity~\omegac{G}, defined to be the size of the largest non-$c$-nilpotent subset of~$G$. In the case that~$L$ is a finite group of Lie type, we identify covers of~$L$ by $c$-nilpotent subgroups, and we use these covers to construct large non-$c$-nilpotent sets in~$L$. We prove that for groups $L$ of fixed rank $r$, there exist constants $D_r$ and $E_r$ such that $D_r N \leq \omega_\infty(L) \leq E_r N$, where $N$ is the number of maximal tori in $L$. %the ambient algebraic group which are stable under the Frobenius endomorphism associated with $L$. In the case of groups~$L$ with twisted rank~$1$, we provide exact formulae for~\omegac{L} for all c\in\longintegers. If we write $q$ for the level of the Frobenius endomorphism associated with $L$ and assume that $q>5$, then these formulae may be expressed as polynomials in $q$ with coefficients in $\mathbb{Z}[\frac12]$

Dirichlet fundamental domains and complex-projective varieties

We prove that for every finitely-presented group G there exists a 2-dimensional irreducible complex-projective variety W with the fundamental group G, so that all singularities of W are normal crossings and Whitney umbrellas.Comment: 1 figur