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cardinality less cf(K cf(X chain of subgroups Chapter characterization of groups class of free class of groups construct a group Corollary 3.2 cyclic groups respectively direct sum direct summand Doctor of Philosophy Eklof elements embedded equivalent to free factor of F free abelian group free algebra free K-free-factor free K-generated K-free free K-generated K-free-factor free product freely group G group iff group of cardinality group of rank group theory groups is definable Hence iff every K-generated isomorphic K-generated subgroup K-pure absolute retract K-pure subgroup language of group least ordinal Lemma Let F limit ordinal Lm equivalent monomorphisms non-free group normal subgroup oooo product of cyclic properties read this thesis regular cardinal sentence which says set of free set of sentences smooth chain stationary strongly compact cardinal subalgebra subgroup is contained subgroup of G sum of cyclic Theorem 2.1 unary relation v e E cf(v weakly compact X-compact Y+l Y+l