On generalized autocommutativity degree of finite groups

Year 2019, Volume: 48 Issue: 2, 472 - 478, 01.04.2019

Parama Dutta Rajat Kanti Nath

Abstract

Let $H$  be  a subgroup of a finite group $G$ and Aut$(G)$ be the automorphism group of $G$. In this paper we introduce and study the probability  that the autocommutator of a randomly chosen pair of elements, one from $H$ and the other from Aut$(G)$, is equal to a given element of  $G$.

Keywords

automorphism group , autocommutativity degree , autoisoclinism

References

• H. Arora and R. Karan, What is the probability an automorphism fixes a group ele- ment?, Comm. Algebra 45 (3), 1141-1150, 2017.
• P. Dutta and R.K. Nath, On relative autocommutativity degree of a subgroup of a finite group, arXiv:1706.05614v1 [math.GR], 2017.
• P. Dutta and R.K. Nath, Autocommuting probabilty of a finite group, Comm. Algebra 46 (3), 961-969, 2018.
• P. Hall, The classification of prime power groups, J. Reine Angew. Math. 182, 130-141, 1940.
• P.V. Hegarty, The absolute centre of a group, J. Algebra 169 (3), 929-935, 1994.
• C. J. Hillar and D. L. Rhea, Automorphism of finite abelian groups, Amer. Math. Monthly 114 (10), 917-923, 2007.
• M.R.R. Moghaddam, M.J. Sadeghifard and M. Eshrati, Some properties of autoisoclin- ism of groups, Fifth International group theory conference, Islamic Azad University, Mashhad, Iran, 13-15 March 2013.
• M.R.R. Moghaddam, F. Saeedi and E. Khamseh, The probability of an automorphism fixing a subgroup element of a finite group, Asian-Eur. J. Math. 4 (2), 301308, 2011.
• M.R. Rismanchian and Z. Sepehrizadeh, Autoisoclinism classes and autocommutativity degrees of finite groups, Hacet. J. Math. Stat. 44 (4), 893-899, 2015.
• G.J. Sherman, What is the probability an automorphism fixes a group element?, Amer. Math. Monthly 82, 261-264, 1975.