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Register a new userRegular subgroups of the affine group with no translations
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Given a regular subgroup R of AGL_n(F), one can ask if R contains nontrivial translations. A negative answer to this question was given by Liebeck, Praeger and Saxl for AGL_2(p) (p a prime), AGL_3(p) (p odd) and for AGL_4(2). A positive answer was given by Hegedus for AGL_n(p) when n >= 4 if p is odd and for n=3 or n >= 5 if p=2. A first generalization to finite fields of Hegedus construction was recently obtained by Catino, Colazzo and Stefanelli. In this paper we give examples of such subgroups in AGL_n(F) for any n >= 5 and any field F. For n < 5 we provide necessary and sufficient conditions for their existence, assuming R to be unipotent if char F=0.
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This paper is a new contribution to the study of regular subgroups of the affine group $AGL_n(F)$, for any field $F$. In particular we associate to any partition $lambda eq (1^{n+1})$ of $n+1$ abelian regular subgroups in such a way that different partitions define non-conjugate subgroups. Moreover, we classify the regular subgroups of certain natural types for $nleq 4$. Our classification is equivalent to the classification of split local algebras of dimension $n+1$ over $F$. Our methods, based on classical results of linear algebra, are computer free.
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 $textsf{PSL}(2,mathbb{Z})$.
We show that finitely-generated, purely pseudo-Anosov subgroups of the genus-2 Goeritz group are convex cocompact in the genus-2 mapping class group.
In this paper, important concepts from finite group theory are translated to localities, in particular to linking localities. Here localities are group-like structures associated to fusion systems which were introduced by Chermak. Linking localities (by Chermak also called proper localities) are special kinds of localities which correspond to linking systems. Thus they contain the algebraic information that is needed to study $p$-completed classifying spaces of fusion systems as generalizations of $p$-completed classifying spaces of finite groups. Because of the group-like nature of localities, there is a natural notion of partial normal subgroups. Given a locality $mathcal{L}$ and a partial normal subgroup $mathcal{N}$ of $mathcal{L}$, we show that there is a largest partial normal subgroup $mathcal{N}^perp$ of $mathcal{L}$ which, in a certain sense, commutes elementwise with $mathcal{N}$ and thus morally plays the role of a centralizer of $mathcal{N}$ in $mathcal{L}$. This leads to a nice notion of the generalized Fitting subgroup $F^*(mathcal{L})$ of a linking locality $mathcal{L}$. Building on these results we define and study special kinds of linking localities called regular localities. It turns out that there is a theory of components of regular localities akin to the theory of components of finite groups. The main concepts we introduce and work with in the present paper (in particular $mathcal{N}^perp$ in the special case of linking localities, $F^*(mathcal{L})$, regular localities and components of regular localities) were already introduced and studied in a preprint by Chermak. However, we give a different and self-contained approach to the subject where we reprove Chermaks theorems and also show several new results.
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