$p$ is a prime number. A group $G$ is called an infinite extraspecial $p$-group if
1) it is infinite,
2) every $g\neq 1$ in $G$ has order $p$,
3) its centre $Z(G)$ coincide with $G'$ and is a cyclic group of order $p$.
It is claimed in several papers that the first order theory of such a group is supersimple of $SU$ rank $1$, but I could not find any proof. Would you know any clear reference about that? (or even better : a proof !)
I have heard that such a $G$ should be interpretable in a $F_p$-vector space $V$ equiped with a skew symmetric non-degenerate bilinear form, and that $V$ should be supersimple of rank $1$, but again, I cannot find any reference.