On a theorem of Rhemtulla on polycyclic groups.
Let \(G\) be a polycyclic-by-finite group. If \(\Gamma\) is a subgroup of \(\Aut\,G\) and \(H\) is a subgroup of \(G\), denote by \(L(\Gamma,H)\) the poset of all intersections of subgroups \(H^\gamma\) (\(\gamma\in\Gamma\)). In particular, put \(L(G,H)=L(\text{Inn\,}G,G)\). In 1967 A. H. Rhemtulla proved that in a polycyclic-by-finite group any intersection of conjugates of a subgroup is the intersection of finitely many of the conjugates. Moreover, he showed that for every subgroup \(H\) of a polycyclic-by-finite group \(G\) the poset \(L(G,H)\) satisfies the minimal condition. In the article under review the author generalizes the latter result. The main theorem is the following: Theorem. Let \(H\) be a subgroup of a polycyclic-by-finite group \(G\) and set \(\Gamma=\Aut\,G\). Then there is a faithful \(\Gamma G\)-module \(V\) (\(\Gamma G\) is the holomorph of \(G\)) which is additively free Abelian of finite rank, a \(G\)-submodule \(U\) of \(V\) with \(V/U\) additively free Abelian and an element \(x\) of \(V\) such that \(H=C_G(x\text{\,mod\,}U)=\{g\in G\mid x(g-1)\in U\}\). Corollary. Let \(H\) be a subgroup of a polycyclic-by-finite group \(G\) and set \(\Gamma=\Aut\,G\). Then there is an integer \(d\) such that the chains in the poset \(L(\Gamma,H)\) have length at most \(d\). In particular \(L(\Gamma,H)\) satisfies the minimal condition and the characteristic core \(H_\Gamma\) of \(H\) in \(G\) is the intersection of at most \(d+1\) subgroups \(H^\gamma\) (\(\gamma\in\Gamma\)).
intersections of conjugates of subgroups
scientific article; zbMATH DE number 5697070