© 1993 by London Mathematical Society
© London Mathematical Society, 1993
Locally Homogeneous Actions and Universal Extensions of Groups
Mathematics Department, Michigan State University East Lansing, Michigan 48824, USA
We study further the class of universal locally finite (u.l.f.) M-extensions E of a countable group G in which the centralizer CE(G) is countable. These were introduced in [8]: M 
AutG/Inn G is the fixed l.f. group of outer automorphisms of G realized in E and E/G is locally finite and has the ‘
-injective property’ with respect to finite M-extensions F of G, that is, F/GCp(G) M and F/G is finite. Here we shall prove that for most G, if |M| = N1 and N1 = 2N0 then there are N2 non-isomorphic u.l.f. M-extensions E of G with |CE,(G)| = N0. We also complete the presentation of [8] by showing that every countably infinite l.f. group L has 2N0 locally homogeneous actions (on U(A), the countable u.l.f. central extension of any countable periodic abelian group A [6, 11]) which extend any given action of L on A and such that no infinite subactions of any two of these locally homogeneous actions are equivalent. We also construct 2N0 nonembeddable locally homogeneous actions of many L on U(A) which have 
--equivalent actions on U(A) of power 2M0. Finally we produce 2N1 complete u.l.f. groups each of which has 2N0 inequivalent universal actions on the countable u.l.f. group. (Universal actions were introduced in [8, Theorem 8] and are a categorical type of locally homogeneous action; the method of obtaining complete u.l.f. groups is that of [4].)