Nicola Sambonet
Presenting a finite group by a free product of finite cyclic groups the Hopf formula for the Schur multiplier affords also a covering group, and this has minimal exponent provided that the order of the generators is preserved. This condition corresponds to a covering projection between simplicial complexes, and so a presentation by a Fuchsian group corresponds to a covering projection between compact surfaces.
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A covering of a group is a finite set of proper subgroups whose union is the whole group. A covering is minimal if there is no covering of smaller cardinality, and it is nilpotent if all its members are nilpotent subgroups. We complete a proof that every group that has a nilpotent minimal covering is solvable, starting from the previously known result that a minimal counterexample is an almost simple finite group.
On minimal coverings of groups by proper normalizers
该研究聚焦群论中群的正规化子覆盖问题，分析了不存在正规化子覆盖的素数阶p-群，并以此为基础确定了部分具有幂零正规化子覆盖的非幂零群。
The covering number of a finite group $G$, denoted $σ(G)$, is the smallest positive integer $k$ such that $G$ is a union of $k$ proper subgroups. We calculate $σ(G)$ for a family of primitive groups $G$ with a unique minimal normal subgroup $N$, isomorphic to $A_n^m$ with $n$ divisible by $6$ and $G/N$ cyclic. This is a generalization of a result of E. Swartz concerning the symmetric groups. We also prove an asymptotic result concerning pairwise generation.
Groups having minimal covering number 2 of diagonal type
本文对具有正规2-覆盖且任何真商群均无此覆盖的有限对角型群进行了完整分类，推进了Garonzi和Lucchini关于正规2-覆盖有限群的分类工作。
Low-genus primitive monodromy groups with a nonunique minimal normal subgroup
该研究证明了在低亏养情况下，仅存在一个具有非唯一最小正规子群的单映射群覆盖，并指出在高亏养情况下，当度数足够大时不存在此类覆盖。
In this paper we prove that a set of points $B$ of PG(n,2) is a minimal blocking set if and only if $<B>=PG(d,2)$ with $d$ odd and $B$ is a set of $d+2$ points of $PG(d,2)$ no $d+1$ of them in the same hyperplane. As a corollary to the latter result we show that if $G$ is a finite 2-group and $n$ is a positive integer, then $G$ admits a $\mathfrak{C}_{n+1}$-cover if and only if $n$ is even and $G\cong (C_2)^{n}$, where by a $\mathfrak{C}_m$-cover for a group $H$ we mean a set $\mathcal{C}$ of size $m$ of maximal subgroups of $H$ whose set-theoretic union is the whole $H$ and no proper subset o…
Given a finite non-cyclic group $G$, call $σ(G)$ the least number of proper subgroups of $G$ needed to cover $G$. In this paper we give lower and upper bounds for $σ(G)$ for $G$ a group with a unique minimal normal subgroup $N$ isomorphic to $A_n^m$ where $n \geq 5$ and $G/N$ is cyclic. We also show that $σ(A_5 \wr C_2)=57$.
A classical problem in the literature seeks the minimal number of proper subgroups whose union is a given finite group. A different question, with applications to error-correcting codes and graph colorings, involves covering vector spaces over finite fields by (minimally many) proper subspaces. In this note we cover $R$-modules by proper submodules for commutative rings $R$, thereby subsuming and recovering both cases above. Specifically, we study the smallest cardinal number $\aleph$, possibly infinite, such that a given $R$-module is a union of $\aleph$-many proper submodules. (1) We complet…
2-covering numbers of some finite solvable groups
本文推翻了关于有限可解群2覆盖数的猜想，指出其不满足σ₂(G)=1+q+q²的结论。
A set of proper subgroups is a covering for a group if its union is the whole group. The minimal number of subgroups needed to cover $G$ is called the covering number of $G$, denoted by $σ(G)$. Determining $σ(G)$ is an open problem for many non-solvable groups. For symmetric groups $S_n$, Maróti determined $σ(S_n)$ for odd $n$ with the exception of $n=9$ and gave estimates for $n$ even. In this paper we determine $σ(S_n)$ for $n = 8$, $9$, $10$ and $12$. In addition we find the covering number for the Mathieu group $M_{12}$ and improve an estimate given by Holmes for the Janko group $J_1$.
Covering by Centralizers
本文研究有限群由元素中心化子的覆盖，证明最大和最小中心化子集合覆盖群，并证明非阿贝尔p-群中非平凡中心化子数量模p余1。
If a group $G$ is the union of proper subgroups $H_1, \dots, H_k$, we say that the collection $\{H_1, \dots H_k \}$ is a cover of $G$, and the size of a minimal cover (supposing one exists) is the covering number of $G$, denoted $σ(G)$. Maróti showed that $σ(S_n) = 2^{n-1}$ for $n$ odd and sufficiently large, and he also gave asymptotic bounds for $n$ even. In this paper, we determine the exact value of $σ(S_n)$ when $n$ is divisible by $6$.
The covering number of a group $G$, denoted by $σ(G)$, is the size of a minimal collection of proper subgroups of $G$ whose union is $G$. We investigate which integers are covering numbers of groups. We determine which integers $129$ or smaller are covering numbers, and we determine precisely or bound the covering number of every primitive monolithic group with a degree of primitivity at most $129$ by introducing effective new computational techniques. Furthermore, we prove that, if $\mathscr{F}_1$ is the family of finite groups $G$ such that all proper quotients of $G$ are solvable, then $\ma…
Given a finite non-cyclic group $G$, call $σ(G)$ the smallest number of proper subgroups of $G$ needed to cover $G$. Lucchini and Detomi conjectured that if a nonabelian group $G$ is such that $σ(G) < σ(G/N)$ for every non-trivial normal subgroup $N$ of $G$ then $G$ is \textit{monolithic}, meaning that it admits a unique minimal normal subgroup. In this paper we show how this conjecture can be attacked by the direct study of monolithic groups.
Fixed point ratios, Sylow numbers and coverings of $p$-elements in finite groups
本文借助Burness和Guralnick的工作，利用本原置换群的不动点比，研究了有限群$G$的Sylow $p$-子群数量及$p$-元素集合真子群覆盖的最小规模。
We prove that for any prime $p\geq 3$ the minimal exponential growth rate of the Baumslag-Solitar group $BS(1,p)$ and the lamplighter group $\mathcal{L}_p=(\mathbb{Z}/p\mathbb{Z})\wr \mathbb{Z}$ are equal. We also show that for $p=2$ this claim is not true and the growth rate of $BS(1,2)$ is equal to the positive root of $x^3-x^2-2$, whilst the one of the lamplighter group $\mathcal{L}_2$ is equal to the golden ratio $(1+\sqrt5)/2$. The latter value also serves to show that the lower bound of A.Mann from [Mann, Journal of Algebra 326, no. 1 (2011) 208--217] for the growth rates of non-semidire…
A \emph{finite cover} of a group $G$ is a finite collection $\mathcal{C}$ of proper subgroups of $G$ with the property that $\bigcup \mathcal{C} = G$. A finite group admits a finite cover if and only if it is noncyclic. More generally, it is known that a group admits a finite cover if and only if it has a finite, noncyclic homomorphic image. If $\mathcal{C}$ is a finite cover of a group $G$, and no cover of $G$ with fewer subgroups exists, then $\mathcal{C}$ is said to be a \emph{minimal cover} of $G$, and the cardinality of $\mathcal{C}$ is called the \emph{covering number} of $G$, denoted by…
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