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M. C. Tamburini Bellani

Publications and source records attributed to M. C. Tamburini Bellani.

10 recordsLinked to original sources
arXiv2024

The $(2,3)$-generation of the finite simple odd-dimensional orthogonal groups

The complete classification of the finite simple groups that are $(2,3)$-generated is a problem which is still open only for orthogonal groups. Here, we construct $(2, 3)$-generators for the finite odd-dimensional orthogonal groups $Ω_{2k+1}(q)$, $k\geq 4$. As a byproduct we also obtain $(2,3)$-generators for $Ω_{4k}^+(q)$ with $k\geq 3$ and $q$ odd, and for $Ω_{4k+2}^\pm(q)$ with $k\geq 4$ and $q\equiv \pm 1 \pmod 4$.

math.GR↗
arXiv2022

The (2,3)-generation of the finite 8-dimensional orthogonal groups

We construct $(2,3)$-generators for the finite $8$-dimensional orthogonal groups, proving the following results: the groups $Ω_8^+(q)$ and $PΩ_8^+(q)$ are $(2,3)$-generated if and only if $q\geq 4$; the groups $Ω_8^-(q)$ and $PΩ_8^-(q)$ are $(2,3)$-generated for all $q\geq 2$.

math.GR↗
arXiv2021

On the (2,3)-generation of the finite symplectic groups

This paper is a new important step towards the complete classification of the finite simple groups which are $(2, 3)$-generated. In fact, we prove that the symplectic groups $Sp_{2n}(q)$ are $(2,3)$-generated for all $n\geq 4$. Because of the existing literature, this result implies that the groups $PSp_{2n}(q)$ are $(2,3)$-generated for all $n\geq 2$, with the exception of $PSp_4(2^f)$ and $PSp_4(3^f)$.

math.GR↗
arXiv2019

The $(2,3)$-generation of the finite unitary groups

In this paper we prove that the unitary groups $SU_n(q^2)$ are $(2,3)$-generated for any prime power $q$ and any integer $n\geq 8$. By previous results this implies that, if $n\geq 3$, the groups $SU_n(q^2)$ and $PSU_n(q^2)$ are $(2,3)$-generated, except when $(n,q)\in\{(3,2),(3,3),(3,5),(4,2), (4,3),(5,2)\}$.

math.GR↗
arXiv2017

Regular subgroups of the affine group with no translations

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.

math.GR↗
arXiv2016

More on regular subgroups of the affine group

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 $λ\neq (1^{n+1})$ 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 $n\leq 4$. 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.

math.GR↗
arXiv2015

Scott's formula and Hurwitz groups

This paper continues previous work, based on systematic use of a formula of L. Scott, to detect Hurwitz groups. It closes the problem of determining the finite simple groups contained in $PGL_n(F)$ for $n\leq 7$ which are Hurwitz, where $F$ is an algebraically closed field. For the groups $G_2(q)$, $q\geq 5$, and the Janko groups $J_1$ and $J_2$ it provides explicit $(2,3,7)$-generators.

math.GR↗
arXiv2015

The simple classical groups of dimension less than 6 which are (2,3)-generated

In this paper we determine the classical simple groups of dimension r=3,5 which are (2,3)-generated (the cases r = 2, 4 are known). If r = 3, they are PSL_3(q), q <> 4, and PSU_3(q^2), q^2 <> 9, 25. If r = 5 they are PSL_5(q), for all q, and PSU_5(q^2), q^2 >= 9. Also, the soluble group PSU_3(4) is not (2,3)-generated. We give explicit (2,3)-generators of the linear preimages, in the special linear groups, of the (2,3)-generated simple groups.

math.GR↗
arXiv2011

Uniform (2,k)-generation of the 4-dimensional classical groups

In this paper we study the (2,k)-generation of the finite classical groups SL(4,q), Sp(4,q), SU(4,q^2) and their projective images. Here k is the order of an arbitrary element of SL(2,q), subject to the necessary condition k>= 3. When q is even we allow also k=4.

math.GR↗