The subgroup lattice of polycyclic groups

Abstract

Let G be a polycyclic-by-finite group, and let X be a subgroup of G. It has been proved by Kegel [Math. Ann. 163 (1966), 248–258] that if the image of X is subnormal in every finite quotient of G, then X is actually subnormal in G; while Robinson [Invent. Math. 10 (1970), 38–43] and Wehrfritz [Proc. London Math. Soc. 20:3 (1970), 101–122] proved that a polycyclic-by-finite group is nilpotent provided that all its finite quotients are nilpotent.

Our first main result (Theorem 2.4) shows that every modular subgroup can be similarly recognized by only looking at the finite quotients of polycyclic-by-finite groups. This extends a theorem of Lennox and Wilson [Arch. Math. (Basel) 28 (1977), 113–116] and improves the main result of Musella [Arch. Math. (Basel) 76 (2001), 161–165] — see also Corollary 2.6.

Our second main result (Theorem 2.16) provides a detailed description of uniquely complemented subgroups in infinite polycyclic-by-finite groups. This is the first non-trivial characterization of this type of subgroups in the infinite case (see [18], p.142), and it has some surprising consequences. It shows in fact that in infinite polycyclic-by-finite groups, the neutral subgroups coincide with the join-distributive subgroups (Corollary 2.31), and that the meet-quasi-distributive subgroups coincide with the uniquely complements subgroups (Theorem 2.22); thus, we face one of those rare occasions in which some types of subgroups coincide in the infinite case but they do not coincide in the finite case. Further relevant consequences of this result deal with detailed descriptions of meet-distributive and join-distributive subgroups (Theorems 2.28 and 2.26), and with the possibility of recognizing all the previously mentioned types of subgroup starting from their images in the finite quotients (Corollary 2.17, Corollary 2.21, Theorem 2.23 and Theorem 2.29).

Finally, in the spirit of Baumslag, Cannonito and Miller III [Math. Z. 153 (1977), 117–134], we also provide theoretical algorithms to determine if a given subgroup of a polycyclic-by-finite group is a modular, join-distributive, meet-(quasi-) distributive, uniquely complemented, and neutral (Corollary 2.9 and Theorem 2.32).