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Uri Bader
Uri Bader
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Here is an answer based on the many comments by myself and by Nick Gill. I am assuming $G$ is not in the list: $\text{Sp}(2,\mathbb{F}_2)$, $\text{Sp}(2,\mathbb{F}_3)$, $\text{Sp}(4,\mathbb{F}_2)$. This is

In characteristic 2 the list of symplectic groups whichtransvections are not simple modulo their center. Theiralways involutions, so generation by involution should be checked directly and I haven't done ittransvections implies generation by involutions in all dimensions. In all other cases it is knownBelow I will assume that every proper normal subgroup of $G$ is central, so the question is equivalent to the question of existence of a non-central involution (as the group generated by thesecharacteristic is non-central and normal)not 2.

In dimensiondimensions 2 the determinant is a symplectic form, thus $G$ is conjugated to $\text{SL}_2(F)$. Then it is easy to check that the only involutions in $G$ has a non-central involution iffare $\text{char}(F)= 2$$1$ and $-1$. In particular, $G$ is not generated by involtions. Below I will assume that the dimension is not 2.

In higher dimensionsI claim that $G$ always contains ais generated by involutions. Note first that $G$ has at least one non-central involution. For: for example one can view the form as a direct sum of lower dimensional ones, take 1 on one and -1 on the other. Note also that the group generated by all non-central involution is normal and not central. This group must be $G$, as every proper normal subgroup of $G$ is central. Indeed, this is the case for every symplectic group apart of $\text{Sp}(2,\mathbb{F}_2)$, $\text{Sp}(2,\mathbb{F}_3)$ and $\text{Sp}(4,\mathbb{F}_2)$, and by the assumptions above $G$ is not in this list.

Here is an answer based on the many comments by myself and by Nick Gill. I am assuming $G$ is not in the list: $\text{Sp}(2,\mathbb{F}_2)$, $\text{Sp}(2,\mathbb{F}_3)$, $\text{Sp}(4,\mathbb{F}_2)$. This is the list of symplectic groups which are not simple modulo their center. Their generation by involution should be checked directly and I haven't done it. In all other cases it is known that every proper normal subgroup of $G$ is central, so the question is equivalent to the question of existence of a non-central involution (as the group generated by these is non-central and normal).

In dimension 2 the determinant is a symplectic form, thus $G$ is conjugated to $\text{SL}_2(F)$. Then it is easy to check that $G$ has a non-central involution iff $\text{char}(F)= 2$.

In higher dimensions $G$ always contains a non-central involution. For example one can view the form as a direct sum of lower dimensional ones, take 1 on one and -1 on the other.

Here is an answer based on the many comments by myself and by Nick Gill.

In characteristic 2 the transvections are always involutions, so generation by transvections implies generation by involutions in all dimensions. Below I will assume that the characteristic is not 2.

In dimensions 2 the determinant is a symplectic form, thus $G$ is conjugated to $\text{SL}_2(F)$. Then it is easy to check that the only involutions in $G$ are $1$ and $-1$. In particular, $G$ is not generated by involtions. Below I will assume that the dimension is not 2.

I claim that $G$ is generated by involutions. Note first that $G$ has at least one non-central involution: for example one can view the form as a direct sum of lower dimensional ones, take 1 on one and -1 on the other. Note also that the group generated by all non-central involution is normal and not central. This group must be $G$, as every proper normal subgroup of $G$ is central. Indeed, this is the case for every symplectic group apart of $\text{Sp}(2,\mathbb{F}_2)$, $\text{Sp}(2,\mathbb{F}_3)$ and $\text{Sp}(4,\mathbb{F}_2)$, and by the assumptions above $G$ is not in this list.

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Uri Bader
Uri Bader
  • 11.6k
  • 2
  • 37
  • 60

Here is an answer based on the many comments by myself and by Nick Gill. I am assuming $G$ is not in the list: $\text{Sp}(2,\mathbb{F}_2)$, $\text{Sp}(2,\mathbb{F}_3)$, $\text{Sp}(4,\mathbb{F}_2)$. This is the list of symplectic groups which are not simple modulo their center. Their generation by involution should be checked directly and I haven't done it. In all other cases it is known that every proper normal subgroup of $G$ is central, so the question is equivalent to the question of existence of a non-central involution (as the group generated by these is non-central and normal).

In dimension 2 the determinant is a symplectic form, thus $G$ is conjugated to $\text{SL}_2(F)$. Then it is easy to check that $G$ has a non-central involution iff $\text{char}(F)= 2$.

In higher dimensions $G$ always contains a non-central involution. For example one can view the form as a direct sum of lower dimensional ones, take 1 on one and -1 on the other.