Existentiell abgeschlossene Sylowturmgruppen. (Existentially closed Sylow tower groups)
For a set \(\pi\) of primes we denote by \(ST_{\pi}\) the class of all locally finite \(\pi\)-groups in which there exists a normal series \(\{(S_ p,T_ p)|\) \(p\in \pi \}\) (of an order type \(<\}\) such that the factor group \(S_ p/T_ p\) is a p-group for every \(p\in \pi\). We prove some properties of existentially closed \(ST_{\pi}\)-groups G, in which the subgroups \(S_ p\) and \(T_ p\) possess complements \(S^ *_ p\) and \(T^ *_ p\) in G. The main results are the following: The group \(T^ *_ p\) is an existentially closed \(ST_{\pi _ 1}\)-group, where \(\pi _ 1=\{q\in \pi |\) \(p\leq q\}\), and \(S^ *_ p\) is an existentially closed \(ST_{\pi _ 1\setminus p}\)-group. The groups G, \(S_ p\) and \(T_ p\) are verbally complete. Furthermore the group G possesses a unique chief series. The order type of this series is a dense, linear order without endpoints. If p is a non-maximal prime in \(\pi\), relative to the order \(<\), then each chief factor M/N in G with \(T_ p\subseteq N\subseteq M\subseteq S_ p\) is an infinite, non-central, elementary-abelian p- group. Some of these results we can prove without the additional assumption, that the groups \(S_ p\) and \(T_ p\) possess complements in G.
existentially closed \(ST_{\pi }\)-groups
scientific article; zbMATH DE number 4046019