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 Grothendiec
k 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 subgr
oups. 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 $mathcal{G}(m)$ f
or 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.
Thomas Breuer
,Gerhard Hiss
,Frank Lubeck
.
(2019)
.
"The completion of the $3$-modular character table of the Chevalley group $F_4(2)$ and its covering group"
.
Gerhard Hiss
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