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Register a new userSubgroups of Classical Groups that are Transitive on Subspaces
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For each finite classical group $G$, we classify the subgroups of $G$ which act transitively on a $G$-invariant set of subspaces of the natural module, where the subspaces are either totally isotropic or nondegenerate. Our proof uses the classification of the maximal factorisations of almost simple groups. As a first application of these results we classify all point-transitive subgroups of automorphisms of finite thick generalised quadrangles.
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Building on earlier results for regular maps and for orientably regular chiral maps, we classify the non-abelian finite simple groups arising as automorphism groups of maps in each of the 14 Graver-Watkins classes of edge-transitive maps.
Sela proved every torsion-free one-ended hyperbolic group is coHopfian. We prove that there exist torsion-free one-ended hyperbolic groups that are not commensurably coHopfian. In particular, we show that the fundamental group of every simple surface amalgam is not commensurably coHopfian.
We prove that a group homomorphism $varphicolon Lto G$ from a locally compact Hausdorff group $L$ into a discrete group $G$ either is continuous, or there exists a normal open subgroup $Nsubseteq L$ such that $varphi(N)$ is a torsion group provided that $G$ does not include $mathbb{Q}$ or the $p$-adic integers $mathbb{Z}_p$ or the Prufer $p$-group $mathbb{Z}(p^infty)$ for any prime $p$ as a subgroup, and if the torsion subgroups of $G$ are small in the sense that any torsion subgroup of $G$ is artinian. In particular, if $varphi$ is surjective and $G$ additionaly does not have non-trivial normal torsion subgroups, then $varphi$ is continuous. As an application we obtain results concerning the continuity of group homomorphisms from locally compact Hausdorff groups to many groups from geometric group theory, in particular to automorphism groups of right-angled Artin groups and to Helly groups.
Let $G$ be a finite group and $sigma ={sigma_{i} | iin I}$ some partition of the set of all primes $Bbb{P}$, that is, $sigma ={sigma_{i} | iin I }$, where $Bbb{P}=bigcup_{iin I} sigma_{i}$ and $sigma_{i}cap sigma_{j}= emptyset $ for all $i e j$. We say that $G$ is $sigma$-primary if $G$ is a $sigma _{i}$-group for some $i$. A subgroup $A$ of $G$ is said to be: ${sigma}$-subnormal in $G$ if there is a subgroup chain $A=A_{0} leq A_{1} leq cdots leq A_{n}=G$ such that either $A_{i-1}trianglelefteq A_{i}$ or $A_{i}/(A_{i-1})_{A_{i}}$ is $sigma$-primary for all $i=1, ldots, n$, modular in $G$ if the following conditions hold: (i) $langle X, A cap Z rangle=langle X, A rangle cap Z$ for all $X leq G, Z leq G$ such that $X leq Z$, and (ii) $langle A, Y cap Z rangle=langle A, Y rangle cap Z$ for all $Y leq G, Z leq G$ such that $A leq Z$. In this paper, a subgroup $A$ of $G$ is called $sigma$-quasinormal in $G$ if $L$ is modular and ${sigma}$-subnormal in $G$. We study $sigma$-quasinormal subgroups of $G$. In particular, we prove that if a subgroup $H$ of $G$ is $sigma$-quasinormal in $G$, then for every chief factor $H/K$ of $G$ between $H^{G}$ and $H_{G}$ the semidirect product $(H/K)rtimes (G/C_{G}(H/K))$ is $sigma$-primary.
We provide the first examples of words in the free group of rank 2 which are not proper powers and for which the corresponding word maps are non-surjective on an infinite family of finite non-abelian simple groups.
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