$p$ 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.



On the theory of infinite extraspecial $p$-groups
As Alain pointed out, extraspecial groups are "the same" as vectors spaces over $\mathbb{F\_p}$ equipped with a bi-linear skew-symmetric form. In fact to complete the answer to your question, one can just add that this identification is elementary. More precisely. the theory of an infinite extraspecial group can be elementary interpreted in the theory of a vector space with a skew-symmetric form. Namely, if $G$ is an extraspecial group, and $V$ is the corresponding vector space with form $<.,.>$. Then one interprets $G$ in $V$ as follows. $G$ is the set of all pairs $V\times \mathbb{F\_p}$ with product $(u,a)\*(v,b)=(uv, a+b+< u, v >)$. Therefore every elementary formula $\theta$ in the signature of $G$ can be rewritten as an elementary formula $\tau(\theta)$ in the theory of $(V, <.,.>)$ (the formula $\tau(\theta)$ holds in $V$ iff $\theta$ holds in $G$). Since the elementary theory of the latter is decidable, the elementary theory of $G$ is decidable too. 


