On decomposing an Abelian \(p\)-group under a \(p'\)-operator group (Q2711613)

Let \(A\) be a \(p'\)-group acting on a finite Abelian \(p\)-group \(P\). Let \(\Omega_1(P)={\mathcal U}_1\otimes\cdots\otimes{\mathcal U}_s\), where \({\mathcal U}_i\) are \(A\)-irreducible for \(i=1,\dots,s\). A necessary and sufficient condition for the existence of an \(A\)-indecomposable decomposition of \(P=R_1\otimes\cdots\otimes R_s\) such that \(\Omega_1(R_i)={\mathcal U}_i\) for \(i=1,\dots,s\), is given. This implies Theorem 1 from the article [\textit{Y. Berkovich}, ibid. 5, No. 2, 143-158 (1998; Zbl 0911.20007)].