On the theory of infinite extraspecial $p$-groups

Trying to understand the last sentence (I'm not a model-theorist!): all the examples of infinite extraspecial $p$-groups I could come up with, are Heisenberg groups over an $F_p$-vector space endowed with a non-degenerate skew-symmetric bilinear form...

If I were to prove that any infinite extraspecial $p$-group is of this form, I would define $V$ as the quotient $G/Z(G)$, and the bilinear form $B(X,Y)$ (for $X,Y\in G/Z(G)$) by $B(X,Y)=[x,y]\in Z(G)=F_p$, where $x,y\in G$ are pre-images of $X,Y$ under the quotient map $G\rightarrow G/Z(G)$. This is clearly well-defined, skew-symmetry and non-degeneracy are obvious, and I have no time to check bilinearity, but I think that it can be left as exercise.

EDIT: Bilinearity follows from standard identities on commutators, see e.g. http://en.wikipedia.org/wiki/Commutator

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.