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Products of conjugacy classes of two by two matrices. (English) Zbl 0840.20040
The covering number $\text{cn} (G)$ and the extended covering number $\text{ecn} (G)$ of a simple noncommutative group is the least $k$ such that $\text{cn} (G)\leq k$ if $C^k= G$ for every nontrivial conjugacy class $C$ in $G$, and $\text{ecn} (G)\leq k$ if $C_1\cdots C_k= G$ for any nontrivial conjugacy classes $C_j$ in $G$, respectively. The authors extend these notions to arbitrary groups: Let $C_1, \dots, C_k$ be any conjugacy classes in $G$ such that the normal subgroup generated by each of them contains $[G,G ]$. Then the extended covering number $\text{ecn} (G)$ is the least $k$ such that the product $C_1\cdots C_k$ contains every similarity class $C_0$ with $C_0= C_1 \cdots C_k \bmod [G,G ]$, and the covering number $\text{cn} (G)$ is the least $k$ satisfying this condition for equal classes $C_1= \dots= C_k$. Clearly, $\text{cn} (G)\leq \text{ecn} (G)$. The authors compute the covering number and the extended covering number for the groups $G= \text{PSL}_2 (F)$, $\text{GL}_2 (F)$, $\text{PGL}_2 (F)$, and $\text{SL}_2 (F)$ over an arbitrary field $F$. It turns out, that $\text{cn} (G)$ and $\text{ecn} (G)$ depend on certain properties of the field. So $\text{cn} (G)$ and $\text{ecn} (G)$ may be 2, 3, 4 or 5 depending on whether the field is quadratically closed, a $C_1 (2)$-field, or not a $C_1 (2)$-field.

MSC:
20G15Linear algebraic groups over arbitrary fields
20F05Generators, relations, and presentations of groups
20D60Arithmetic and combinatorial problems on finite groups
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References:
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