# On the theory of infinite extraspecial $p$-groups

$p$ 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.

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Note that for $p=2$ this definition makes $G$ abelian. In finite group theory, where extraspecial groups often arise, one normally does not request that each element has order $p$, instead, a weaker condition, $G/Z(G)$ is elementary abelian $p$-group, is imposed... – Dima Pasechnik Nov 19 '11 at 12:17
[nitpicking]1 has never order p. It has always order 1.[/nitpicking] – Johannes Hahn Nov 19 '11 at 13:42
@Johannes Hahn. :) right ! Corrected ! – Drike Nov 19 '11 at 14:11
@Dima: in your comment, you want to say "because of condition 2) in question, G turns out to be abelian"? I was wondering, whether infinite extra-special p-groups could occur for p=2. – Philip Oct 6 '15 at 4:15
Philip, if $a^2=b^2=(ab)^2=1$ then $ab=ba$, and you get an abelian group. – Dima Pasechnik Oct 7 '15 at 0:10

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.

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@ Drike: Mark's reply is very useful, and I think it answers the comment you made on my answer. A small complement: the group structure Mark describes on $V \times F_p$ is precisely what is often called the Heisenberg group of the pair $(V,<.,.>)$. – Alain Valette Nov 19 '11 at 17:30
@all: cristal clear ! Thanks ! – Drike Nov 19 '11 at 18:47

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

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@Alain Valette. Thanks for the answer. This shows that $G/Z(G)$ can be viewed as a vector-space over Fp equiped with a skew symmetric bilinear form, but I do not see why one can recover all the information about $G$ from this vector space only (using any "natural" operations : [], +, Fp quotients, finite cartesian products etc.) – Drike Nov 19 '11 at 12:55

Supersimplicity of rank 1 of the infinite extra-special p-group follows from Proposition 3.11 and Lemma 4.1 of

D. Macpherson and C. Steinhorns, One dimensional asymptotic classes of finite structures, TAMS 360, 2008.

A different approach can be found in

A. Baudish, Neostability of Fraïssé limits of 2-nilpotent groups of exponent p>2, Archive for Mathematical Logic 55, 397-403, 2016.
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