Neostability-properties of Fraïssé limits of 2-nilpotent groups of exponent \(p>2\)
For a natural number \(p>2\), an infinite group \(G\) is called an \textit{extra-special \(p\)-group} if \(g^p=1\) for every \(g\in G\), if the centre of \(G\) is cyclic of order \(p\) and equals \(G'\) (the first derived subgroup of \(G\)). Infinite extra-special \(p\)-groups are a special case of groups of the same name introduced by \textit{P. Hall} and \textit{G. Higman} [Proc. Lond. Math. Soc. (3) 6, 1--42 (1956; Zbl 0073.25503)]. Their first-order theory has been investigated by \textit{U. Felgner} [Log. Anal., Nouv. Sér. 18, 407--428 (1975; Zbl 0355.02038)] who showed it is complete, \(\aleph_0\)-categorical, but not \(\aleph_1\)-categorical. The only countable such group (up to isomorphism) is thus commonly referred to as \textit{the infinite extra-special \(p\)-group}. Following the author, we write it \(D_1\). In the paper under review, the author introduces for every natural numbers \(p>2\) and \(n>0\) a group called \(D_n\) constructed as the \textit{Fraïssé} limit of the class \(\mathcal K_n\) of those finite groups of exponent \(p\) whose derived subgroups are central and generated by no more than \(n\) elements. The \textit{Fraïssé} limit of \(\mathcal K_1\) is Felgner's countable extra special \(p\)-group. The author shows: {\parindent=0.7cm \begin{itemize}\item[(1)] that the class \(\mathcal K_n\) indeed has a Fraïssé limit, \(D_n\); \item[(2)] that the first-order theory of \(D_n\) is \(\aleph_0\)-categorical and has full elimination of quantifiers (Theorem 1); \item[(3)] that \(D_n\) has Shelah's independance property (Theorem 2); \item[(4)] that \(D_n\) is supersimple of SU-rank 1 (Theorem 3). \end{itemize}} See also \textit{S. Shelah} and \textit{J. Steprāns}' [Ann. Pure Appl. Logic 34, 87--97 (1987; Zbl 0667.04002)] for a further studies of the categoricity of the theory of \(D_1\) in higher powers, as well as \textit{D. Macpherson} and \textit{C. Steinhorn} [Trans. Am. Math. Soc. 360, No. 1, 411--448 (2008; Zbl 1127.03026)], from which follows implicitly that \(D_1\) is supersimple of SU-rank 1.
scientific article; zbMATH DE number 6580186
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