Centraliser dimension and universal classes of groups. (English) Zbl 1118.20030

Duncan, A. J.; Kazachkov, I. V.; Remeslennikov, V. N.

Sib. Èlektron. Mat. Izv. 3, 197-215 (2006).

Summary: We prove results that will be required for the study of the algebraic geometry of partially commutative groups. We define classes of groups axiomatized by sentences determined by a graph. Among the classes which arise this way we find CSA- and CT-groups. We study the centralisers of a group, with particular attention to the height of the lattice of centralisers, which we call the centraliser dimension of the group. The behaviour of centraliser dimension under several common group operations is described. Groups with centraliser dimension 2 are studied in detail. It is shown that CT-groups are precisely those with centraliser dimension 2 and trivial centre.

MSC:

20E15	Chains and lattices of subgroups, subnormal subgroups
20E10	Quasivarieties and varieties of groups
14A22	Noncommutative algebraic geometry
20F10	Word problems, other decision problems, connections with logic and automata (group-theoretic aspects)
03B25	Decidability of theories and sets of sentences

Keywords:

commutativity; commutative transitivity; conjugacy-separable groups; malnormal subgroups; algebraic geometry of partially commutative groups; classes of groups; lattices of centralisers