Let $q$ be odd, $G=PSU\_n(q)$ (Projective Special Unitary group) and $H=PSU\_{n-1}(q)$. Is it always true that $H$ is a subgroup of $G\ ?$



question about projective special unitary group
${\rm PSU}\_5(3)$ (which is isomorphic to ${\rm SU}\_5(3)$) has a subgroup isomorphic to ${\rm SU}\_4(3)$ (which has centre of order $4$), but none isomorphic to ${\rm PSU}\_4(3)$.


In general ${\rm SU}\_n(q)$ contains ${\rm SU}\_{n-1}(q)$, but factoring out the centre of the first of these, which has order $(n,q+1)$, does not always result in factoring out the centre of the second, which has order $(n-1,q+1)$.


