On autocommutators and the autocommutator subgroup in infinite abelian groups
-
Luise-Charlotte Kappe
and
Mercede Maj
Abstract
It is well known that the set of commutators in a group usually does not form a subgroup. A similar phenomenon occurs for the set of autocommutators. There exists a group of order 64 and nilpotency class 2, where the set of autocommutators does not form a subgroup, and this group is of minimal order with this property. However, for finite abelian groups, the set of autocommutators is always a subgroup. We will show in this paper that this is no longer true for infinite abelian groups. We characterize finitely generated infinite abelian groups in which the set of autocommutators does not form a subgroup and show that in an infinite abelian torsion group the set of commutators is a subgroup. Lastly, we investigate torsion-free abelian groups with finite automorphism group and we study whether the set of autocommutators forms a subgroup in those groups.
Dedicated to the memory of Wolfgang Kappe
Funding statement: This work was supported by the “National Group for Algebraic and Geometric Structures, and their Applications” (INdAM – GNSAGA), Italy.
Acknowledgements
The first author is grateful to INdAM – GNSAGA for support and to the Department of Mathematics of the University of Salerno for hospitality while this investigation was carried out. We would like to thank the anonymous referee of this paper for many helpful comments and useful suggestions.
References
[1] R. Baer, Abelian groups without elements of finite order, Duke Math. J. 3 (1937), 68–122. 10.1215/S0012-7094-37-00308-9Search in Google Scholar
[2] M. Bruckheimer, A. C. Bryan and A. Muir, Groups which are the union of three subgroups, Amer. Math. Monthly 77 (1970), no. 1, 52–57. 10.1080/00029890.1970.11992416Search in Google Scholar
[3]
A. L. S. Corner,
Groups of units of orders in
[4] A. de Vries and A. B. de Miranda, Groups with a small number of automorphisms, Math. Z. 68 (1958), 450–464. 10.1007/BF01160361Search in Google Scholar
[5] H. Dietrich and P. Moravec, On the autocommutator subgroup and absolute centre of a group, J. Algebra 341 (2011), 150–157. 10.1016/j.jalgebra.2011.05.038Search in Google Scholar
[6] L. Fuchs, Abelian Groups, Pergamon Press, Oxford, 1960. Search in Google Scholar
[7] L. Fuchs, Infinite Abelian Groups, Vol. II, Academic Press, New York, 1973. Search in Google Scholar
[8] L. Fuchs, Abelian Groups, Springer Monogr. Math., Springer, Cham, 2015. 10.1007/978-3-319-19422-6Search in Google Scholar
[9] D. Garrison, L.-C. Kappe and D. Yull, Autocommutators and the autocommutator subgroup, Combinatorial Group Theory, Discrete Groups, and Number Theory, Contemp. Math. 421, American Mathematical Society, Providence (2006), 137–146. 10.1090/conm/421/08033Search in Google Scholar
[10] J. T. Hallet and K. A. Hirsch, Torsion-free groups having finite automorphism group, J. Algebra 2 (1965), 287–298. 10.1016/0021-8693(65)90010-4Search in Google Scholar
[11] J. T. Hallet and K. A. Hirsch, Die Konstruktion von Gruppen mit vorgeschriebenen Automorphismengruppen, J. Reine Angew. Math. 239–240 (1969), 32–46. 10.1515/crll.1969.239-240.32Search in Google Scholar
[12] P. V. Hegarty, Autocommutator subgroups of finite groups, J. Algebra 190 (1997), 556–562. 10.1006/jabr.1996.6921Search in Google Scholar
[13] D. J. S. Robinson, A Course in the Theory of Groups, Springer, New York, 1995. Search in Google Scholar
[14] G. Scorza, I gruppi che possono pensarsi come somma di tre sottogruppi, Boll. Unione Mat. Ital. 5 (1926), 216–218. Search in Google Scholar
© 2018 Walter de Gruyter GmbH, Berlin/Boston
Articles in the same Issue
- Frontmatter
- Distinct distances between points and lines in 𝔽q2
- A global perspective to connections on principal 2-bundles
- Integral group rings of solvable groups with trivial central units
- Lefschetz property and powers of linear forms in 𝕂[x,y,z]
- On quasi-convex null sequences in infinite cyclic groups
- On autocommutators and the autocommutator subgroup in infinite abelian groups
- Anticyclotomic p-ordinary Iwasawa theory of elliptic modular forms
- Finite-dimensional representations of Leavitt path algebras
- Orlicz dual affine quermassintegrals
- Universal locally finite maximally homogeneous semigroups and inverse semigroups
- Von Neumann algebras, L-algebras, Baer *-monoids, and Garside groups
- Boundedness and compactness of Hardy-type integral operators on Lorentz-type spaces
- A shifted convolution sum for \mathrm{GL}(3) × \mathrm{GL}(2)
- Equivariant Gauss sum of finite quadratic forms
- Right 2-Engel elements, central automorphisms and commuting automorphisms of Lie algebras
-
Solution of Brauer’s
Articles in the same Issue
- Frontmatter
- Distinct distances between points and lines in 𝔽q2
- A global perspective to connections on principal 2-bundles
- Integral group rings of solvable groups with trivial central units
- Lefschetz property and powers of linear forms in 𝕂[x,y,z]
- On quasi-convex null sequences in infinite cyclic groups
- On autocommutators and the autocommutator subgroup in infinite abelian groups
- Anticyclotomic p-ordinary Iwasawa theory of elliptic modular forms
- Finite-dimensional representations of Leavitt path algebras
- Orlicz dual affine quermassintegrals
- Universal locally finite maximally homogeneous semigroups and inverse semigroups
- Von Neumann algebras, L-algebras, Baer *-monoids, and Garside groups
- Boundedness and compactness of Hardy-type integral operators on Lorentz-type spaces
- A shifted convolution sum for \mathrm{GL}(3) × \mathrm{GL}(2)
- Equivariant Gauss sum of finite quadratic forms
- Right 2-Engel elements, central automorphisms and commuting automorphisms of Lie algebras
-
Solution of Brauer’s
