$nX$-Complementary Generations of the Chevalley Group $G_{2}(3)$
نویسندگان
1 School of Mathematical and Computer Sciences, University of Limpopo (Turfloop), P. Bag X1106, Sovenga 0727, South Africa
2 School of Mathematical and Computer Sciences, University of Limpopo (Turfloop), P. Bag X1106, Sovenga 0727, South Africa
3 School of Mathematical and Computer Sciences, University of Limpopo (Turfloop), P. Bag X1106, Sovenga 0727, South Africa
4 School of Mathematical and Statistical Sciences, PAA Focus Area, North-West University (Mahikeng), P. Bag X2046, Mmabatho 2790, South Africa
doi
10.22052/mir.2024.254730.1460چکیده
A finite non-abelian group $G$ is said to be $(l,m,n)$-generated if it can be generated by two elements $x$ and $y$ such that $o(x)=l$, $o(y)=m$ and $o(xy)=n$. Also, $G$ is said to be $nX$-complementary generated if given an arbitrary non-identity element $x\in G$, there exists an element $y \in nX$ such that $G=\langle x,y\rangle$. We studied the $(p,q,r)$-generation for the Chevalley group $G_{2}(3)$, where $p$, $q$ and $r$ are all the primes dividing the order of $G_{2}(3)$. In the current paper, we classify all the non-trivial conjugacy classes of $G_{2}(3)$ whether they are complementary generators or not. To achieve this, we mainly used the structure constant method together with other results applied to establish generation and non-generation of the group $G_{2}(3)$ by the $(p,q,r)$ triples. Some particular algorithms, as well as the (Gap) programming tool, and the Atlas of finite groups have been exploited in our computations.
