Abstract
A group G is an A-group if x α x = xx α for all \({x\in G}\) and all automorphisms α of G. Such groups have nilpotency class at most 3; we construct the first example having class precisely 3.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Abdollahi A., Faghihi A., Mohammadi Hassanabadi A.: 3-generator groups whose elements commute with their endomorphic images are abelian. Comm. Algebra 36, 3783–3791 (2008)
Abdollahi A., Faghihi A., Mohammadi Hassanabadi A.: Minimal number of generators and minimum order of a non-abelian group whose elements commute with their endomorphic images. Comm. Algebra 36, 1976–1987 (2008)
Bosma W., Cannon J., Playoust C.: The Magma algebra system I: The user language. J. Symbolic Comput. 24, 235–265 (1997)
Brooksbank P.A., O’Brien E.A.: Constructing the group preserving a system of forms. Internat. J. Algebra Comput. 18, 227–241 (2008)
E. Costi, Constructive membership testing in classical groups, PhD thesis, Queen Mary, University of London, 2009.
Eick B., Leedham-Green C.R., O’Brien E.A.: Constructing the automorphism group of a p-group. Comm. Algebra 30, 2271–2295 (2002)
Faudree R.: Groups in which each element commutes with its endomorphic images. Proc. Amer. Math. Soc. 27, 236–240 (1971)
The GAP Group, GAP – Groups, Algorithms, and Programming, Version 4.4.12; 2008. http://www.gap-system.org.
Holt D.F., Eick B., O’Brien E.A.: Handbook of computational group theory. Chapman and Hall/CRC, London (2005)
The Kourovka Notebook, Unsolved problems in group theory, 16th ed., Novosibirsk, 2006.
Malone J.J.: More on groups in which each element commutes with its endomorphic images. Proc. Amer. Math. Soc. 65, 209–214 (1977)
Newman M.F., O’Brien E.A.: Application of computers to questions like those of Burnside. II. Internat, J. Algebra Comput. 6, 593–605 (1996)
Nickel W.: Computation of nilpotent Engel groups. J. Austral. Math. Soc. Ser. A 67, 214–222 (1999)
Pettet M.R.: On automorphisms of A-groups. Arch. Math. (Basel) 91, 289–299 (2008)
Robinson D.J.S.: A course in the theory of groups, 2nd ed. Springer-Verlag, New York (1995)
Traustason G.: Symplectic alternating algebras. Internat. J. Algebra Comput. 18, 719–757 (2008)
Author information
Authors and Affiliations
Corresponding author
Additional information
We thank Max Neunhöffer for helpful discussions. The work of Abdollahi was partially supported by the Center of Excellence for Mathematics, University of Isfahan. The work of O’Brien was partially supported by the Marsden Fund of New Zealand via grant UOA721.
Rights and permissions
About this article
Cite this article
Abdollahi, A., Faghihi, A., Linton, S.A. et al. Finite 3-groups of class 3 whose elements commute with their automorphic images. Arch. Math. 95, 1–7 (2010). https://doi.org/10.1007/s00013-010-0144-y
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00013-010-0144-y