Let $G$ denote the projective special linear group $\text{PSL}(2,q)$, for a prime power $q$. It is shown that a finite 2-subgroup of the group $V(\mathbb{Z}G)$ of augmentation 1 units in the integral group ring $\mathbb{Z}G$ of $G$ is isomorphic to a subgroup of $G$. Furthermore, it is shown that a composition factor of a finite subgroup of $V(\mathbb{Z}G)$ is isomorphic to a subgroup of $G$.
Factorization of groups into Zappa-Szep product, or more generally into k-fold Zappa-Szep product of its subgroups, is an interesting problem, since it eases the multiplication of two elements in a group, and has recently been applied for public-key cryptography as well. We give a generalization of the k-fold Zappa-Szep product of cyclic groups, which we call OGS decomposition. It is easy to see that existence of an OGS decomposition for all the composition factors of a non-abelian group G implies the existence of an OGS for G itself. Since the composition factors of a soluble group are cyclic…
For $O$ an imaginary quadratic ring, we compute a fundamental polyhedron of $\text{PE}_2(O)$, the projective elementary subgroup of $\text{PSL}_2(O)$. This allows for new, simplified proofs of theorems of Cohn, Nica, Fine, and Frohman. Namely, we obtain a presentation for $\text{PE}_2(O)$, show that it has infinite-index and is its own normalizer in $\text{PSL}_2(O)$, and split $\text{PSL}_2(O)$ into a free product with amalgamation that has $\text{PE}_2(O)$ as one of its factors.
Let $p$ be a prime number and $q=p^m$, with $m \geq 1$ if $p \neq 2,3$ and $m>1$ otherwise. Let $Ω$ be any non-trivial twist for the complex group algebra of $\mathbf{PSL}_2(q)$ arising from a $2$-cocycle on an abelian subgroup of $\mathbf{PSL}_2(q)$. We show that the twisted Hopf algebra $(\mathbb{C} \mathbf{PSL}_2(q))_Ω$ does not admit a Hopf order over any number ring. The same conclusion is proved for the Suzuki groups, and for $\mathbf{SL}_3(p)$ when the twist stems from an abelian $p$-subgroup. This supplies new families of complex semisimple (and simple) Hopf algebras that do not admit …
The Bianchi groups ${\rm Bi}(d)={\rm PSL}(2,\mathcal{O}_d) < {\rm PSL}(2,\C)$ (where $\mathcal{O}_d$ denotes the ring of integers of $\Q (i\sqrt{d})$, with $d \geqslant 1$ squarefree) can be viewed as subgroups of ${\rm SO}(3,1)$ under the isomorphism ${\rm PSL}(2,\C) \simeq {\rm SO}^0(3,1)$. We study the deformations of these groups into the larger Lie groups ${\rm SU}(3,1)$ and ${\rm SL}(4,\R)$ for small values of $d$. In particular we show that ${\rm Bi}(3)$, which is rigid in ${\rm SO}(3,1)$, admits a 1-dimensional deformation space into ${\rm SU}(3,1)$ and ${\rm SL}(4,\R)$, whereas any de…
We study the Prime Graph Question for integral group rings. This question can be reduced to almost simple groups by a result of Kimmerle and Konovalov. We prove that the Prime Graph Question has an affirmative answer for all almost simple groups having a socle isomorphic to $\operatorname{PSL}(2, p^f)$ for $f \leq 2$, establishing the Prime Graph Question for the first time for all automorphic extensions of series of simple groups. Using this, we determine exactly how far the so-called HeLP-method can take us for (almost simple) groups having an order divisible by at most $4$ different primes.
Let $ G $ be a finite non-solvable group with a primitive irreducible character $ χ$ that vanishes on one conjugacy class. We show that $ G $ has a homomorphic image that is either almost simple or a Frobenius group. We also classify such groups $ G $ with a composition factor isomorphic to a sporadic group, an alternating group $ \rm{A}_{n} $, $ n\geq 5 $ or $ \rm{PSL}_{2}(q) $, where $ q\geq 4 $ is a prime power, when $ χ$ is faithful. Our results partially answer a question of Dixon and Rahnamai Barghi.
Università degli Studi di Modena e Reggio Emilia（摩德纳和雷焦艾米利亚大学）；University of Southern California（南加州大学）；University of Lincoln（林肯大学）；University of Brasilia（巴西利亚大学）
