Die Automorphismengruppe der abzählbaren, existentiell abgeschlossenen \({\mathcal P}_ A\)-Gruppe \(E_ A\). (The automorphism group of the countable existentially closed \({\mathcal P}_ A\)-group \(E_ A)\) (Q1093013)

For a fixed prime p and a countable abelian p-group A let \({\mathcal P}_ A\) be the class of all locally finite p-groups P, in which A is a subgroup of the centre of P. By \textit{F. Leinen} and \textit{R. E. Phillips} [Math. Proc. Camb. Philos. Soc. 100, 281-301 (1986; Zbl 0613.20017)] there exists (up to isomorphism) a unique countable existentially closed \({\mathcal P}_ A\)-group \(E_ A\). In the present paper the structure of the automorphism group \(Aut(E_ A)\) of the group \(E_ A\) is given. This generalizes \textit{F. Leinen}'s description of the structure of the group \(Aut(E_{\{1\}})\) [J. Algebra 103, 160-183 (1986; Zbl 0598.20037)]. The main result is, that the factor group \(Aut(E_ A)/LokInn(E_ A)\) is isomorphic to the direct product of the groups Aut(A) and \(Aut(E_{\{1\}})/LokInn(E_{\{1\}})\), where LokInn(G) means the set of all locally inner automorphisms of the group G.