Correction of a theorem on the symmetric group generated by transvections

Abstract

Let $V$ denote a vector space over two-element field ${\mathbb{F}}_2$ with finite positive dimension and endowed with a symplectic form $B.$ Let $\mathrm{SL}(V)$ denote the special linear group of $V.$ Let $S$ denote a subset of $V.$ Define $Tv(S)$ as the subgroup of $\mathrm{SL}(V)$ generated by the transvections with direction $\alpha$ for all $\alpha \in S.$ Define $G(S)$ as the graph whose vertex set is $S$ and where $\alpha ,\beta \in S$ are connected whenever $B(\alpha ,\beta )=1.$ A well-known theorem states that under the assumption that $S$ spans $V,$ the following (i), (ii) are equivalent: (i) $Tv(S)$ is isomorphic to a symmetric group. (ii) $G(S)$ is a claw-free block graph. We give an example which shows that this theorem is not true. We give a modification of this theorem as follows. Assume that $S$ is a linearly independent set of $V$ and no element of $S$ is in the radical of $V.$ Then the above (i), (ii) are equivalent.

7 pages