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Oliver
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Let $G$ be a group. An involution is an element $g\in G$ such that $g^2=1$. Let $F$ be a filedfield, $V$ an $F$-vector space and $b:V\times V \rightarrow F$ a nondegenerate alternating bilinear form. The set $\mathrm{Sp}(V,b)=\{ f:V\rightarrow V \mid f \mathrm{\ is\ bijective\ and\ } b(f(x),f(y))=b(x,y) \forall x,y \in V\}$ with the composition of maps is a group called the symplectic group of $(V,b)$. It is well-known that the symplectic group is generated by transvections. Is it also generated by involutions in $\mathrm{Sp}(V,b)$?

Let $G$ be a group. An involution is an element $g\in G$ such that $g^2=1$. Let $F$ be a filed, $V$ an $F$-vector space and $b:V\times V \rightarrow F$ a nondegenerate alternating bilinear form. The set $\mathrm{Sp}(V,b)=\{ f:V\rightarrow V \mid f \mathrm{\ is\ bijective\ and\ } b(f(x),f(y))=b(x,y) \forall x,y \in V\}$ with the composition of maps is a group called the symplectic group of $(V,b)$. It is well-known that the symplectic group is generated by transvections. Is it also generated by involutions in $\mathrm{Sp}(V,b)$?

Let $G$ be a group. An involution is an element $g\in G$ such that $g^2=1$. Let $F$ be a field, $V$ an $F$-vector space and $b:V\times V \rightarrow F$ a nondegenerate alternating bilinear form. The set $\mathrm{Sp}(V,b)=\{ f:V\rightarrow V \mid f \mathrm{\ is\ bijective\ and\ } b(f(x),f(y))=b(x,y) \forall x,y \in V\}$ with the composition of maps is a group called the symplectic group of $(V,b)$. It is well-known that the symplectic group is generated by transvections. Is it also generated by involutions in $\mathrm{Sp}(V,b)$?

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Oliver
Oliver
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Let $G$ be a group. An involution is an element $g\in G$ such that $g^2=1$. Let $F$ be a filed, $V$ an $F$-vector space and $b:V\times V \rightarrow F$ ana nondegenerate alternating bilinear form. The set $\mathrm{Sp}(V,b)=\{ f:V\rightarrow V \mid f \mathrm{\ is\ bijective\ and\ } b(f(x),f(y))=b(x,y) \forall x,y \in V\}$ with the composition of maps is a group called the symplectic group of $(V,b)$. It is well-known that the symplectic group is generated by transvections. Is it also generated by involutions in $\mathrm{Sp}(V,b)$?

Let $G$ be a group. An involution is an element $g\in G$ such that $g^2=1$. Let $F$ be a filed, $V$ an $F$-vector space and $b:V\times V \rightarrow F$ an alternating bilinear form. The set $\mathrm{Sp}(V,b)=\{ f:V\rightarrow V \mid f \mathrm{\ is\ bijective\ and\ } b(f(x),f(y))=b(x,y) \forall x,y \in V\}$ with the composition of maps is a group called the symplectic group of $(V,b)$. It is well-known that the symplectic group is generated by transvections. Is it also generated by involutions in $\mathrm{Sp}(V,b)$?

Let $G$ be a group. An involution is an element $g\in G$ such that $g^2=1$. Let $F$ be a filed, $V$ an $F$-vector space and $b:V\times V \rightarrow F$ a nondegenerate alternating bilinear form. The set $\mathrm{Sp}(V,b)=\{ f:V\rightarrow V \mid f \mathrm{\ is\ bijective\ and\ } b(f(x),f(y))=b(x,y) \forall x,y \in V\}$ with the composition of maps is a group called the symplectic group of $(V,b)$. It is well-known that the symplectic group is generated by transvections. Is it also generated by involutions in $\mathrm{Sp}(V,b)$?

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Oliver
Oliver
  • 367
  • 2
  • 7

Generation of the symplectic by involutions

Let $G$ be a group. An involution is an element $g\in G$ such that $g^2=1$. Let $F$ be a filed, $V$ an $F$-vector space and $b:V\times V \rightarrow F$ an alternating bilinear form. The set $\mathrm{Sp}(V,b)=\{ f:V\rightarrow V \mid f \mathrm{\ is\ bijective\ and\ } b(f(x),f(y))=b(x,y) \forall x,y \in V\}$ with the composition of maps is a group called the symplectic group of $(V,b)$. It is well-known that the symplectic group is generated by transvections. Is it also generated by involutions in $\mathrm{Sp}(V,b)$?