Abstract
Let G be a group. An automorphism \(\alpha \) of G is called a commuting automorphism if \(\alpha (x)x= x \alpha (x)\) for all \(x \in G\). The set of all commuting automorphisms of G is denoted by A(G). The set A(G) does not necessarily form a subgroup of the automorphism group of G. If A(G) form a subgroup, then we say G is an A-group. In this paper, we show that the direct product of two finite A-groups is also an A-group. We also show that GL(n, q) for \(n = 3\) or \(q >n\), PSL(2, q) and ZM-groups are A-groups.
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Abdollahi, A., Akbari, S., Maimani, H.R.: Non-commuting graph of a group. J. Algebra 298, 468–492 (2006)
Azad, A., Iranmanesh, M.A., Praeger, C.E., Spiga, P.: Abelian covering of finite general linear groups and an application to their non-commuting graphs. J. Algebra Comb. 34, 683–710 (2011)
Azad, A., Praeger, C.E.: Maximal subsets of pairwise non-commuting elements of three-dimensional general linear groups. Bull. Aust. Math. Soc. 80, 91–104 (2009)
Bell, H.E., Martindale III, W.S.: Centralizing mappings of semiprime rings. Can. Math. Bull. 30, 92–101 (1987)
Bidwell, J.N.S., Curran, M.J., McCaughan, D.J.: Automorphisms of direct products of finite groups. Arch. Math. (Basel) 86, 481–489 (2006)
Deaconescu, M., Silberberg, G., Walls, G.L.: On commuting automorphisms of groups. Arch. Math. (Basel) 79, 423–429 (2002)
Divinsky, N.: On commuting automorphisms of rings. Trans. Roy. Soc. Can. Sect. III (3) 49, 19–22 (1955)
Fouladi, S., Orfi, R.: Commuting automorphisms of some finite groups. Glas. Mat. Ser. III 48(68), 91–96 (2013)
Herstein, I.N.: Problems and solutions: elementary problems: E3039. Am. Math. Mon. 91, 203 (1984)
Huppert, B.: Endliche Gruppen I. Springer, Berlin (1967)
Laffey, T.J.: Problems and solutions: solutions of elementary problems: E3039. Am. Math. Mon. 93, 816–817 (1986)
Luh, J.: A note on commuting automorphisms of rings. Am. Math. Mon. 77, 61–62 (1970)
Neumann, P.M., Praeger, C.E.: Cyclic matrices over finite fields. J. Lond. Math. Soc. 52, 263–284 (1995)
Rai, P.K.: On commuting automorphisms of finite \(p\)-groups. Proc. Japan Acad. Ser. A 91, 57–60 (2015)
Vosooghpour, F., Akhavan-Malayeri, M.: On commuting automorphisms of \(p\)-groups. Commun. Algebra 41, 1292–1299 (2013)
Zassenhaus, H.: Theory of Groups. Chelsea, New York (1949)
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Communicated by F. Degiovanni.
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Kumar, P. On commuting automorphisms of finite groups. Ricerche mat 68, 899–904 (2019). https://doi.org/10.1007/s11587-019-00444-0
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DOI: https://doi.org/10.1007/s11587-019-00444-0