On small Sylow numbers of finite groups
Let $G$ be a finite group and $n_p(G)$ the number of Sylow $p$-subgroups of $G$. In this paper, we prove if $n_p(G)<p^2$ then almost all numbers $n_p(G)$ are a power of a prime.
math.GR↗
arXiv subjects
Igor Lima
Publications and source records attributed to Igor Lima.
Subgroup Separability of Artin Groups II
In this paper, we establish a new criterion for determining whether an Artin group is subgroup separable (LERF), building upon a criterion introduced in a previous work. Specifically, we prove an Artin group is LERF if and only if it does not contain certain induced subgraphs, thus providing a more direct generalization of the Metaftsis-Raptis criterion for RAAGs. As consequences, we prove an Artin group is ERF if and only if it is a free abelian group and we establish a link between subgroup separability and coherence for Artin groups.
math.GR↗
Subgroup separability of surface braid groups and virtual braid groups
In this paper, we study subgroup separability (also known as LERF) and related properties for surface braid groups and virtual (singular) braid groups.
math.GR↗
On pro-p Cappitt groups with finite exponent
A pro-p Cappitt group is a pro-p group G such that the subgroup topologically generated by all non-normal closed subgroups is a proper subgroup of G. In this paper we prove that non-abelian pro-p Cappitt groups whose torsion subgroup is closed has finite exponent. We also prove that in a pro-p Cappitt group its subgroup commutator is a procyclic central subgroup. Finally we show that pro-2 Cappitt groups of exponent 4 are pro-2 Dedekind groups. These results are pro-p versions of the generalized Dedekind groups studied by Cappitt.
math.GR↗
Exposing Bugs in JavaScript Engines through Test Transplantation and Differential Testing
Context. JavaScript is a popular programming language today with several implementations competing for market dominance. Although a specification document and a conformance test suite exist to guide engine development, bugs occur and have important practical consequences. Implementing correct engines is challenging because the spec is intentionally incomplete and evolves frequently. Objective. This paper investigates the use of test transplantation and differential testing for revealing functional bugs in JavaScript engines. The former technique runs the regression test suite of a given engine on another engine. The latter technique fuzzes existing inputs and then compares the output produced by different engines with a differential oracle. Method. We conducted experiments with engines from five major players-Apple, Facebook, Google, Microsoft, and Mozilla-to assess the effectiveness of test transplantation and differential testing. Results. Our results indicate that both techniques revealed several bugs, many of which confirmed by developers. We reported 35 bugs with test transplantation (23 of these bugs confirmed and 19 fixed) and reported 24 bugs with differential testing (17 of these confirmed and 10 fixed). Results indicate that most of these bugs affected two engines-Apple's JSC and Microsoft's ChakraCore (24 and 26 bugs, respectively). To summarize, our results show that test transplantation and differential testing are easy to apply and very effective in finding bugs in complex software, such as JavaScript engines.
cs.SE↗
Subgroup Separability of Artin Groups
We find a condition on the underlying graph of an Artin group that fully determines if it is subgroup separable. As a consequence, an Artin group is subgroup separable if and only if it can be obtained from Artin groups of ranks at most 2 via a finite sequence of free products and direct products with the infinite cyclic group. This result generalizes the Metaftsis-Raptis criterion for Right-Angled Artin groups.
math.GR↗
A new characterization of simple $K_3$-groups using same-order type
Let $G$ be a group, define an equivalence relation $\sim$ as below: $$\forall \ g, h \in G, \ g \sim h \Longleftrightarrow |g| = |h|$$ the set of sizes of equivalence classes with respect to this relation is called the same-order type of $G$ and denoted by $\alpha(G)$. And $G$ is said a $\alpha_n$-group if $|\alpha(G)|=n$. Let $\pi(G)$ be the set of prime divisors of the order of $G$. A simple group of $G$ is called a simple $K_n$-group if $|\pi(G)|=n$. We give a new characterization of simple $K_3$-groups using same-order type. Indeed we prove that a nonabelian simple group $G$ has same-order type \{r, m, n, k, l\} if and only if $G \cong PSL(2,q)$, with $q=7, 8$ or $9$. This result generalizes the main results in \cite{KKA}, \cite{Sh} and \cite{TZ1}. Motived by the main result in \cite{TZ1} L. J. Taghvasani and M. Zarrin put the following Conjecture 2.10: \textit{Let $S$ be a nonabelian simple $\alpha_n$-group and $G$ a $\alpha_n$-group such that $|S|=|G|$. Then $S \cong G$}. In this paper with a counterexample we give a negative answer to this question.
math.GR↗
Maximal covers of finite groups
Let $\lambda(G)$ be the maximum number of subgroups in an irredundant covering of the finite group $G$. We prove that if $G$ is a group with $\lambda(G) \leqslant 6$, then $G$ is supersolvable. We also describe the structure of the groups $G$ with $\lambda(G)=6$. Moreover, we show that if $G$ is a group with $\lambda(G) \leqslant 30$, then $G$ is solvable.
math.GR↗
On the number of cyclic subgroups of a finite group
Let $G$ be a finite group and let $c(G)$ be the number of cyclic subgroups of $G$. We study the function $\alpha(G) = c(G)/|G|$. We explore its basic properties and we point out a connection with the probability of commutation. For many families $\mathscr{F}$ of groups we characterize the groups $G \in \mathscr{F}$ for which $\alpha(G)$ is maximal and we classify the groups $G$ for which $\alpha(G) > 3/4$. We also study the number of cyclic subgroups of a direct power of a given group deducing an asymptotic result and we characterize the equality $\alpha(G) = \alpha(G/N)$ when $G/N$ is a symmetric group.
math.GR↗