Abstract
Let \( G \) be a group and let \( \operatorname{Aut}_{c}(G) \) be the group of the class preserving automorphisms of \( G \). We prove the following: (i) If \( G \) is (nilpotent of class \( c \))-by-(soluble of derived length \( d \)), then \( \operatorname{Aut}_{c}(G) \) is (nilpotent of class \( \leq c-1 \))-by-(soluble of derived length \( d+1 \) or \( d \)), which extends a result of Rai. (ii) If \( G \) is a \( B_{1} \)-group, then \( \operatorname{Aut}_{c}(G) \) is (nilpotent of class \( \leq n-1 \))-by-soluble, where \( n \) is the length of a finite chain of \( G \).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Burnside W., Theory of Groups of Finite Order. Second Edition, Dover, New York (1911).
Burnside W., “On the outer automorphisms of a group,” Proc. London Math. Soc., vol. 11, 40–42 (1913).
Malinowska I., “On quasi-inner automorphisms of a finite \( p \)-group,” Publ. Math. Debrecen, vol. 41, no. 2, 73–77 (1992).
Wall G. E., “Finite groups with class-preserving outer automorphisms,” J. London Math. Soc., vol. 22, no. 4, 315–320 (1947).
Yadav M. K., “On automorphisms of some finite \( p \)-groups,” Proc. Indian Acad. Sci., vol. 118, no. 1, 1–11 (2008).
Endimioni G., “Pointwise inner automorphisms in a free nilpotent group,” Quart. J. Math., vol. 53, no. 4, 397–402 (2002).
Feit W. and Seitz G. M., “On finite rational groups and related topics,” Illinois J. Math., vol. 33, no. 1, 103–131 (1988).
Hertweck M. and Jespers E., “Class-preserving automorphisms and normalizer property for Blackburn groups,” J. Group Theory, vol. 12, no. 1, 157–169 (2009).
Herman A. and Li Y., “Class preserving automorphisms of Blackburn groups,” J. Aust. Math. Soc., vol. 80, no. 3, 351–358 (2006).
Ono T. and Wada H., “Hasse principle for symmetric and alternating groups,” Proc. Japan Acad., vol. 75, no. 4, 61–62 (1999).
Bardakov V., Vesnin A., and Yadav M. K., “Class preserving automorphisms of unitriangular groups,” Internat. J. Algebra Comput., vol. 22, no. 3, Article no. 1250023 (2012).
Hertweck M., “Class-preserving automorphisms of finite groups,” J. Algebra, vol. 241, no. 1, 1–26 (2001).
Yadav M. K., “Class preserving automorphisms of finite \( p \)-groups,” J. London Math. Soc., vol. 75, no. 3, 755–772 (2007).
Sah C. H., “Automorphisms of finite groups,” J. Algebra, vol. 10, no. 1, 47–68 (1968).
Rai P. K., “On class preserving automorphisms of groups,” Ric. Mat., vol. 63, no. 2, 189–194 (2014).
Lennox J. C. and Robinson D. J. S., The Theory of Infinite Soluble Groups, Oxford University, Oxford (2004).
Robinson D. J. S., A Course in the Theory of Groups. Second Edition, Springer, New York (1996).
Segal D., Polycyclic Groups, Cambridge University, Cambridge (1983).
Fuchs L., Infinite Abelian Groups. Vol. 2, Academic, New York and London (1973).
Endimioni G., “Automorphisms fixing every normal subgroup of a nilpotent-by-abelian group,” Rend. Sem. Mat. Univ. Padova, vol. 120, 73–77 (2008).
Acknowledgments
We thank the referees for their time and comments.
Funding
The project was supported by the National Natural Science Foundation of China (Grants nos. 11801129 and 12171142) and the Hebei Provincial Natural Science Foundation (Grant no. A2019402211).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Sibirskii Matematicheskii Zhurnal, 2022, Vol. 63, No. 3, pp. 639–644. https://doi.org/10.33048/smzh.2022.63.312
Rights and permissions
About this article
Cite this article
Xu, T., Liu, H. Class Preserving Automorphisms of Groups. Sib Math J 63, 530–534 (2022). https://doi.org/10.1134/S0037446622030120
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0037446622030120