Commutator endomorphisms of totally projective abelian 𝑝-groups
-
Patrick Keef
Abstract
For a primary abelian group 𝐺, Chekhlov and Danchev (2015) defined three variations of Kaplansky’s notion of full transitivity by restricting one’s attention to the subgroup, the subring and the unitary subring of the endomorphism ring of 𝐺 generated by the collection of all commutator endomorphisms. They posed the problem of describing exactly which totally projective groups exhibit these forms of full transitivity. This problem, and some closely related questions, are completely answered using the Ulm function of 𝐺.
Acknowledgements
The author expresses his thanks to the referee who made suggestions that significantly improved the exposition in this work.
-
Communicated by: John S. Wilson
References
[1] A. R. Chekhlov and P. V. Danchev, On commutator fully transitive Abelian groups, J. Group Theory 18 (2015), no. 4, 623–647. 10.1515/jgth-2015-0014Suche in Google Scholar
[2] A. R. Chekhlov and P. V. Danchev, On commutator Krylov transitive and commutator weakly transitive Abelian p-groups, Forum Math. 31 (2019), no. 6, 1607–1623. 10.1515/forum-2019-0066Suche in Google Scholar
[3] L. Fuchs, Infinite Abelian Groups. Vol. II, Pure Appl. Math. Vol. 36-II, Academic Press, New York, 1973. Suche in Google Scholar
[4] P. Hill, On transitive and fully transitive primary groups, Proc. Amer. Math. Soc. 22 (1969), 414–417. 10.1090/S0002-9939-1969-0269735-0Suche in Google Scholar
[5] R. Hunter, F. Richman and E. Walker, Existence theorems for Warfield groups, Trans. Amer. Math. Soc. 235 (1978), 345–362. 10.1090/S0002-9947-1978-0473044-4Suche in Google Scholar
[6] I. Kaplansky, Infinite abelian groups, University of Michigan Press, Ann Arbor, 1954. Suche in Google Scholar
[7] Z. Mesyan, Commutator rings, Bull. Austral. Math. Soc. 74 (2006), no. 2, 279–288. 10.1017/S0004972700035711Suche in Google Scholar
© 2023 Walter de Gruyter GmbH, Berlin/Boston
Artikel in diesem Heft
- Frontmatter
- The Higman operations and embeddings of recursive groups
- Redundant relators in cyclic presentations of groups
- Commutator endomorphisms of totally projective abelian 𝑝-groups
- Algebraic groups over finite fields: Connections between subgroups and isogenies
- Relative stable equivalences of Morita type for the principal blocks of finite groups and relative Brauer indecomposability
- 5-Regular prime graphs of finite nonsolvable groups
- On weak commutativity in 𝑝-groups
- More on chiral polytopes of type \{4, 4, …, 4\} with~solvable automorphism groups
Artikel in diesem Heft
- Frontmatter
- The Higman operations and embeddings of recursive groups
- Redundant relators in cyclic presentations of groups
- Commutator endomorphisms of totally projective abelian 𝑝-groups
- Algebraic groups over finite fields: Connections between subgroups and isogenies
- Relative stable equivalences of Morita type for the principal blocks of finite groups and relative Brauer indecomposability
- 5-Regular prime graphs of finite nonsolvable groups
- On weak commutativity in 𝑝-groups
- More on chiral polytopes of type \{4, 4, …, 4\} with~solvable automorphism groups
