The Independence of Kaplansky's Notions of Transitivity and Full Transitivity

On projection-invariant subgroups of Abelian 𝑝-groups

“…In either case P π ≤ P and so P is a projection-invariant subgroup of C. However direct calculation shows that P [2] = P is not any of the subgroups (2 ω C) [2], (2 ω+1 C) [2], (2 ω+2 C) [2], (2 ω+3 C) [2] = 0. Since a ∈ P [2] has height ω in C, P [2] = (p n C) [2] for any finite n. Thus C is not projectively socle-regular but it is strongly socle-regular since it is transitive -see [4, Theorem 2.4].…”

“…Consider any of the groups C constructed by Corner in [2] which were transitive but not fully transitive; these groups had the property that It is shown in [8,Example 3.16] that the elements of Φ can be described by two families {θ iλ } and {φ jµ } with the parameters 1 ≤ i, j ≤ 4 and λ ∈ {±1, ±3}, µ ∈ {0, ±1, 2}. A straightforward check using the definitions given in Example 3.16 of [8], reveals that the only idempotent endomorphisms in Φ are 0 and 1.…”

Fully transitive $p$-groups with finite first Ulm subgroup

“…In [5] Files and Goldsmith proved the surprising result that a p-group G is fully transitive if and only if its square G ⊕ G is transitive. Nevertheless, for p = 2 the independence of both concepts was shown by Corner in [4] and already Kaplansky had shown in [11] that for p > 2 transitivity always implies full transitivity. Therefore it is natural to ask which fully transitive non-transitive p-groups appear.…”

“…By a fundamental observation of Corner [4] one can reduce the decision of whether or not a p-group G is (fully) transitive to its first Ulm subgroup p ω G. In [4] it was shown that G is (fully) transitive if and only if End(G) p ω G acts (fully) transitively on the first Ulm subgroup p ω G of G, i.e., for any x, y ∈ p ω G such that U p ω G (x) = U p ω G (y) (U p ω G (x) ≤ U p ω G (y)) there exists an automorphism (endomorphism) α of G such that α(x) = y. Corner constructed a fully transitive non-transitive p-group with countable first Ulm subgroup in [4] and it is a longstanding problem whether there exists a fully transitive non-transitive p-group with finite first Ulm subgroup. Partial results were obtained by Carroll and Goldsmith in [2], [3] and by Hennecke in [9], but a general solution hasn't been found yet.…”

Torsion-Free Weakly Transitive Abelian Groups

“…Since then the notions have been extensively studied and the concepts have been widened to include torsion-free and mixed abelian groups and modules see e.g. Carroll and Goldsmith (1996), Corner (1976), Files and Goldsmith (1998), Grinshpon (1982), Hennecke (1999), Hill (1969), Megibben (1966), Paras and Strüngmann (2003) for the torsion case, Files (1996Files ( , 1997, Hennecke and Strüngmann (2000) for the mixed case and Dobrusin (1985), Dugas and Shelah (1989), Grinshpon (1982), Mathematics Subject Classification (2000): 20K01, 20K10, 20K30. * Supported by the Deutsche Forschungs Gesellschaft.…”