  For $p$ an odd prime and $F$ the cyclic group of order $p$, we show that the
number of conjugacy classes of embeddings of $F$ in $SU(p)$ such that no
element of $F$ has 1 as an eigenvalue is $(1+C_{p-1})/p$, where $C_{p-1}$ is a
Catalan number. We prove that the only coset space $SU(p)/F$ that admits a
$p$-local $H$-structure is the classical Lie group $PSU(p)$. We also show that
$SU(4)/\mathbb Z_3$, where $\mathbb Z_3$ is embedded off the center of $SU(4)$,
is a novel example of an $H$-space, even globally. We apply our results to the
study of homotopy classes of maps from $BF$ to $BSU(n)$.
