High subgroups of primary Abelian groups of length \(\omega +n\)
The main goal of this paper is the reduction of the study of abelian p- groups of length \(\omega +n\) having all high subgroups isomorphic, to the study of groups G of length \(\omega\) having a specified set of pure dense subgroups isomorphic. All groups in this paper are abelian p-groups. Let \(\alpha\) be a limit ordinal and n a positive integer. For \(i=1,2\) let \(G_ i\) be a p-group of length \(\alpha\) and \(P_ i\) a subgroup of \(G_ i[p^ n]\) containing a maximal \(p^{n-1}\)-bounded summand of \(G_ i\) and dense in the \(\alpha\)-topology (i.e., topology on \(G_ i\) having \(\{p^{\beta}G_ i\}_{\beta<\alpha}\) as a base for the neighborhoods of \(\{\) 0\(\})\). \((G_ 1,P_ 1)\) and \((G_ 2,P_ 2)\) are said to be equivalent if there is an isomorphism \(\Psi\) from \(G_ 1\) onto \(G_ 2\) such that \(\Psi(P_ 1)=P_ 2.\) If \(\alpha =\omega\), such a pair (G,P) is said to be of IH-type if given any two dense pure subgroups \(H_ 1\) and \(H_ 2\) of G such that \(P=H_ 1[p^ n]\oplus Q_ 1=H_ 2[p^ n]\oplus Q_ 2\) with \(p^{n-1}Q_ 1=p^{n-1}Q_ 2=\{0\},\) we have \(H_ 1\cong H_ 2.\) Let \({\mathfrak G}_{\alpha,n}\) be the class of equivalence classes of pairs (G,P), \(P[p]\neq G[p],\) and \({\mathfrak A}_{\alpha +n}\) the class of isomorphism classes of p-groups of length \(\alpha +n\). Then the map \(\phi:{\mathfrak A}_{\alpha +n}\to {\mathfrak G}_{\alpha,n}\) defined by \(\phi(A)=(A/p^{\alpha}A,A[p^ n]/p^{\alpha}A)\) is one-to-one and onto. Let \(\hat {\mathfrak A}_{\omega +n}\) be the subclass of \({\mathfrak A}_{\omega +n}\) of IH-groups, and \(\hat {\mathfrak G}_{\omega,n}\) the subclass of \({\mathfrak G}_{\omega,n}\) of IH-types. Then \(\phi| \hat {\mathfrak A}_{\omega +n}\) is onto \(\hat {\mathfrak G}_{\omega,n}\). An example is constructed showing that for any integer \(n\geq 2\) there exists a p-group A such that all the high subgroups of A are isomorphic but not all the high subgroups of \(A/p^{\omega +n-1}A\) are isomorphic.
scientific article; zbMATH DE number 3851381