LF groups, aec amalgamation, few automorphisms

by Shelah. [Sh:1098]

In section 1 we deal with amalgamation bases, e.g. we define when an a.e.c. k has (lambda, kappa)-amalgamation which means ``many'' M in K^k_lambda are amalgamation bases. We then consider what occurs for the class of lf groups. In section 2 we deal with weak definability of a in N backslash M over M, for bold K_{{exlf}} . In section 3 we deal with indecomposable members of K_exlf and with the existence of a universal members of K^k_mu, for mu strong limit of cofinality aleph_0 . Most note worthy: if bold K_lf has a universal model in mu then it has a canonical one similar to the special models, (the parallel to saturated ones in their cardinality). In section 4 we prove ``every G in bold K^lf_{<= mu} can be extended to a complete (lambda, theta)-full G'' for many cardinals; we may consider fixing the outer auto group. In section(0C) we solve a problem of Heckin on the automorphism group of the Hall group.

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