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Automorphism group of the commuting graph of $ 2\times 2 $ matrix ring over $ \mathbb{Z}_{p^{s}} $

  • Received: 30 June 2021 Accepted: 27 August 2021 Published: 02 September 2021
  • MSC : 20B25, 15B33

  • Let $ R $ be a ring with identity. The commuting graph of $ R $ is the graph associated to $ R $ whose vertices are non-central elements in $ R $, and distinct vertices $ A $ and $ B $ are adjacent if and only if $ AB = BA $. In this paper, we completely determine the automorphism group of the commuting graph of $ 2\times 2 $ matrix ring over $ \mathbb{Z}_{p^{s}} $, where $ \mathbb{Z}_{p^{s}} $ is the ring of integers modulo $ p^{s} $, $ p $ is a prime and $ s $ is a positive integer.

    Citation: Hengbin Zhang. Automorphism group of the commuting graph of $ 2\times 2 $ matrix ring over $ \mathbb{Z}_{p^{s}} $[J]. AIMS Mathematics, 2021, 6(11): 12650-12659. doi: 10.3934/math.2021729

    Related Papers:

  • Let $ R $ be a ring with identity. The commuting graph of $ R $ is the graph associated to $ R $ whose vertices are non-central elements in $ R $, and distinct vertices $ A $ and $ B $ are adjacent if and only if $ AB = BA $. In this paper, we completely determine the automorphism group of the commuting graph of $ 2\times 2 $ matrix ring over $ \mathbb{Z}_{p^{s}} $, where $ \mathbb{Z}_{p^{s}} $ is the ring of integers modulo $ p^{s} $, $ p $ is a prime and $ s $ is a positive integer.



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    [1] A. Abdollahi, Commuting graphs of full matrix rings over finite fields, Linear Algebra Appl., 428 (2008), 2947–2954.
    [2] A. Mohammadian, On commuting graphs of finite matrix rings, Commun. Algebra, 38 (2010), 988–994.
    [3] S. Akbari, H. Bidkhori, A. Mohammadian, Commuting graphs of matrix algebras, Commun. Algebra, 36 (2008), 4020–4031.
    [4] D. Bundy, The connectivity of commuting graphs, J. Comb. Theory Ser. A, 113 (2006), 995–1007.
    [5] M. Herzog, P. Longobardi, M. Maj, On a commuting graph on conjugacy classes of groups, Commun. Algebra, 37 (2009), 3369–3387.
    [6] M. Mirzargar, P. P. Pach, A. R. Ashrafi, The automorphism group of commuting graph of a finite group, Bull. Korean Math. Soc., 51 (2014), 1145–1153.
    [7] M. Mirzargar, P. P. Pach, A. R. Ashrafi, Remarks on commuting graph of a finite group, Electron. Notes Discrete Math., 45 (2014), 103–106.
    [8] J. Zhou, Automorphisms of the commuting graph over $2\times2$ matrix ring, Acta Sci. Nat. Univ. Sunyatseni, 55 (2016), 39–43.
    [9] B. R. McDonald, Finite rings with identity, New York: Marcel Dekker, Inc., 1974.
    [10] J. J. Rotman, An introduction to the theory of groups, 4 Eds., New York: Springer-Verlag, 1995.
    [11] T. Ceccherini-Silberstein, F. Scarabotti, F. Tolli, Representation theory and harmonic analysis of wreath products of finite groups, Cambridge University Press, 2014.
    [12] M. D. Neusel, L. Smith, Invariant theory of finite groups, American Mathematical Society, 2001.
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  • © 2021 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
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