### $1{\textstyle \frac{1}{2}}$-generation of finite simple groups.

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2010 Mathematics Subject Classification: 20F05, 20D06.We prove that the group PSL6(q) is (2,3)-generated for any q. In fact, we provide explicit generators x and y of orders 2 and 3, respectively, for the group SL6(q).

We present a short and self-contained proof of the extension property for partial isometries of the class of all finite metric spaces.

We prove that an Artin-Tits group of type $\tilde{C}$ is the group of fractions of a Garside monoid, analogous to the known dual monoids associated with Artin-Tits groups of spherical type and obtained by the “generated group” method. This answers, in this particular case, a general question on Artin-Tits groups, gives a new presentation of an Artin-Tits group of type $\tilde{C}$, and has consequences for the word problem, the computation of some centralizers or the triviality of the center. A key point of the proof...

$G(3,m,n)$ is the group presented by $\langle a,b\mid {a}^{5}={\left(ab\right)}^{2}={b}^{m+3}{a}^{-n}{b}^{m}{a}^{-n}=1\rangle $. In this paper, we study the structure of $G(3,m,n)$. We also give a new efficient presentation for the Projective Special Linear group $PSL(2,5)$ and in particular we prove that $PSL(2,5)$ is isomorphic to $G(3,m,n)$ under certain conditions.