Abstract
The aim of this paper is to state some results on an \(\alpha \)-nilpotent group, which was recently introduced by Barzegar and Erfanian (Caspian J. Math. Sci. 4(2) (2015) 271–283), for any fixed automorphism \(\alpha \) of a group G. We define an identity nilpotent group and classify all finitely generated identity nilpotent groups. Moreover, we prove a theorem on a generalization of the converse of the known Schur’s theorem. In the last section of the paper, we study absolute normal subgroups of a finite group.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Abdollahi A, Finite \(p\)-groups of class \(2\) have non-inner automorphisms of order \(p\), J. Algebra, 312 (2007) 876–879
Barzegar R and Erfanian A, Nilpotency and solubility of groups relative to an automorphism, Caspian J. Math. Sci., 4(2) (2015) 271–283
Christopher J H and Darren L R, Automorphism of finite abelian groups, Am. Math. Mon., 114(10) (2007) 917–923
Deaconescu M and Silberberg G, Non-inner automorphisms of order \(p\) of finite \(p\)-groups, J. Algebra, 250 (2002) 283–287
Ganjali M and Erfanian A, Perfect groups and normal subgroups related to an automorphism, Ricerche Mat. 66 (2017) 407–413
Garrison D, Kappe L-C and Yull D, Autocommutators and the autocommutator subgroup, Contemp. Math., 421 (2006) 137–146
Gaschütz W, Nichtabelsche \(p\)-gruppen besitzen äussere \(p\)-automorphismen, J. Algebra, 4 (1966) 1–2
Hegarty P, The absolute centre of a group, J. Algebra, 169 (1994) 929–935
Khukhro E I, Nilpotent groups and their automorphisms (1993) (Berlin: Walter de Gruyter)
Miller G A, Determination of all the groups of order \(p^{m}\) which contain the abelian group of type \((m-2, 1)\), \(p\) being any prime, Tran. AMS, 2(3) (1901) 259–272
Miller G A, On the groups of order \(p^m\) which contain operators of order \(p^{m-2}\), Trans. AMS, 3(4) (1902) 383–387
Niroomand P, The converse of Schur’s theorem, Arch. Math., 94(5) (2010) 401–403
Parvaneh F, Some properties of autonilpotent groups, Italian J. Pure Appl. Math., 35 (2015) 1–8
Robinson D J S, A course in the theory of groups, Graduate Texts in Mathematics 80 (1996) (New York: Springer)
Winter D L, The automorphism group of an extraspecial \(p\)-group, Rocky Mountain J. Math., 2(2) (1972) 159–168
Acknowledgements
The authors would like to thank the referee for helpful suggestions. This research was supported by a Grant from Ferdowsi University of Mashhad-Graduate Studies (No. 29078).
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicating Editor: B Sury
Rights and permissions
About this article
Cite this article
Erfanian, A., Ganjali, M. Nilpotent groups related to an automorphism. Proc Math Sci 128, 60 (2018). https://doi.org/10.1007/s12044-018-0441-0
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s12044-018-0441-0