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Register a new userThe completion of the $3$-modular character table of the Chevalley group $F_4(2)$ and its covering group
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Using computational methods, we complete the determination of the $3$-modular character table of the Chevalley group $F_4(2)$ and its covering group.
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Let $U$ be a Sylow $p$-subgroup of the finite Chevalley group of type $D_4$ over the field of $q$ elements, where $q$ is a power of a prime $p$. We describe a construction of the generic character table of $U$.
We count the finitely generated subgroups of the modular group $textsf{PSL}(2,mathbb{Z})$. More precisely: each such subgroup $H$ can be represented by its Stallings graph $Gamma(H)$, we consider the number of vertices of $Gamma(H)$ to be the size of $H$ and we count the subgroups of size $n$. Since an index $n$ subgroup has size $n$, our results generalize the known results on the enumeration of the finite index subgroups of $textsf{PSL}(2,mathbb{Z})$. We give asymptotic equivalents for the number of finitely generated subgroups of $textsf{PSL}(2,mathbb{Z})$, as well as of the number of finite index subgroups, free subgroups and free finite index subgroups. We also give the expected value of the isomorphism type of a size $n$ subgroup and prove a large deviations statement concerning this value. Similar results are proved for finite index and for free subgroups. Finally, we show how to efficiently generate uniformly at random a size $n$ subgroup (resp. finite index subgroup, free subgroup) of $textsf{PSL}(2,mathbb{Z})$.
In this article, we discuss the Grothendieck group completion (GGC) of a gyrogroup. Consequently, we show that there is a one to one correspondence between actions and representations of a gyrogroup, and actions and representations of its Grothendieck group completion. We also introduce the concept of an action of a right gyrogroup.
We show how to count and randomly generate finitely generated subgroups of the modular group $textsf{PSL}(2,mathbb{Z})$ of a given isomorphism type. We also prove that almost malnormality and non-parabolicity are negligible properties for these subgroups. The combinatorial methods developed to achieve these results bring to light a natural map, which associates with any finitely generated subgroup of $textsf{PSL}(2,mathbb{Z})$ a graph which we call its silhouette, and which can be interpreted as a conjugacy class of free finite index subgroups of $textsf{PSL}(2,mathbb{Z})$.
Let $(mathcal{G},Gamma)$ be an abstract graph of finite groups. If $Gamma$ is finite, we can construct a profinite graph of groups in a natural way $(hat{mathcal{G}},Gamma)$, where $hat{mathcal{G}}(m)$ is the profinite completion of $mathcal{G}(m)$ for all $m in Gamma$. The main reason for this is that $Gamma$ is finite, so it is already profinite. In this paper we deal with the infinite case, by constructing a profinite graph $overline{Gamma}$ where $Gamma$ is densely embedded and then defining a profinite graph of groups $(widehat{mathcal{G}},overline{Gamma})$. We also prove that the fundamental group $Pi_1(widehat{mathcal{G}},overline{Gamma})$ is the profinite completion of $Pi_1^{abs}(mathcal{G},Gamma)$. This answers Open Question 6.7.1 of the book Profinite Graphs and Groups, published by Luis Ribes in 2017. Later we generalise the main theorem of a paper by Luis Ribes and the second author, proving that if $R$ is a virtually free abstract group and $H$ is a finitely generated subgroup of $R$, then $overline{N_{R}(H)}=N_{hat{R}}(overline{H})$ answering Open Question 15.11.10 of the book of Ribes. Finally, we generalise the main theorem of a paper by Sheila Chagas and the second author, showing that every virtually free group is subgroup conjugacy separable. This answers Open Question 15.11.11 of the same book of Ribes.
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