Some results on perfect groups

Document Type : Original Paper

Authors

1 Department of Mathematics‎, ‎Faculty of Science‎, ‎Razi University‎, ‎Kermanshah‎, ‎Iran

2 Department of Mathematic, Faculty of Mathematical Sciences, University of Science and Technology of Mazandaran, Behshahr, Iran.

Abstract

Let G be a perfect group. In this paper we use a new method to prove that any automorphism of G can be lifted to a unique automorphism of its covering group. Also, we show that if G is a central factor of some group H then any automorphism of G can be lifted to aunique homomorphism from the covering group of G to H.

Keywords

Main Subjects


[1] Holt, D.F. and Plesken, W. (1989). Perfect groups, Clarendon Press, Oxford.
[2] Arias, D., Casas, J.M. and Ladra, M. (2007). On universal central extensions of precrossed and crossed modules, J. Pure Appl. Algebra, 210, 177–191.
[3] Casas, J.M. and Van der Linden, T. (2014). Universal central extensions in semi-abelian categories, Appl. Category Struct., 22(1), 253–268.
[4] Lassueur, C. and Thévenaz, J. (2017). Universal p′-central extensions, Expositiones Mathematicae, 35, 237-251. [5] Donadze, G., Ladra, M. and Thomas, V. (2017). On some closure properties of the non-abelian tensor product, J. Algebra, 427, 399-413.
[6] Brown, R. and Loday, J.L. (1987). Van Kampen theorems for diagrams of spaces, Topology, 26, 311-335.
[7] Brown, R., Johnson, D.L. and Robertson, E.F. (1987). Some computations of nonabelian tensor products of groups, J. Algebra, 111, 177–202. [8] Karpilovsky, G. (1987). The Schur Multiplier, Clarendon Press, Oxford, UK.
[9] Alperin, J.L. and Gorenstein, D. (1966). The multiplicators of certain simple groups, Proc. Amer. Math. Soc., 17, 515-519.
[10] Moghaddam, M.R.R. and Salemkar, A.R. (2000).Varietal isologism and covering groups, Arch. Math. (Basel), 75, 8-15.
[11] Brown, R., Johnson, D.L. and Robertson, E.F. (1987). Some computations of nonabelian tensor products of groups, J. Algebra, 111, 177–202.
[12] Thompson, J.G. (1973). Isomorphisms induced by automorphisms, J. Austral. Math. Soc., 16, 16-17.