Abstract
If \(A\) is a nontrivial torsion-free, locally cyclic group with no nontrivial divisible quotients, and \(G\) is the split extension of \(A\) by a group of order 2 acting on \(A\) by means of the inverting map, then \(G\simeq {{{\mathrm{Aut}}}G} \). We prove that in no other case the full automorphism group of a group is infinite and locally dihedral.
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Cutolo, G., Smith, H., Wiegold, J.: \(p\)-groups of maximal class as automorphism groups. Ill. J. Math. 47(1–2), 141–156 (2003)
Cutolo, G., Smith, H., Wiegold, J.: Wreath products of cyclic \(p\)-groups as automorphism groups. J. Algebra 282(2), 610–625 (2004)
Dixon, M.R., Evans, M.J.: Periodic divisible-by-finite automorphism groups are finite. J. Algebra 137(2), 416–424 (1991)
Evans, M.J.: Freely decomposable automorphism groups. Arch. Math. (Basel) 52(5), 420–423 (1989)
Fournelle, T.: Finite groups of automorphisms of infinite groups. I. J. Algebra 70(1), 16–22 (1981)
Fournelle, T.A.: Finite groups of automorphisms of infinite groups. II. J. Algebra 80(1), 106–112 (1983)
Franciosi, S., de Giovanni, F.: A note on groups with countable automorphism groups. Arch. Math. (Basel) 47(1), 12–16 (1986)
Fuchs, L.: Infinite abelian groups, vol. II. Pure and applied mathematics, vol. 36-II, Academic Press, New York (1973)
de Giovanni, F., Russo, A.: The infinite dihedral group as automorphism group. Ricerche Mat. 51(2), 337–339 (2002)
Iyer, H.K.: On solving the equation \({\rm Aut}(X)=G\). Rocky mountain. J. Math. 9(4), 653–670 (1979)
Lennox, J.C., Robinson, D.J.S.: The theory of infinite soluble groups. Oxford mathematical monographs. The Clarendon Press Oxford University Press, Oxford (2004)
Pettet, M.R.: Locally finite groups as automorphism groups. Arch. Math. (Basel) 48(1), 1–9 (1987)
Robinson, D.J.S.: Infinite torsion groups as automorphism groups. Q. J. Math. 30(119), 351–364 (1979)
Robinson, D.J.S.: A course in the theory of group. Graduate Texts in Mathematics, vol. 80, 2nd edn. Springer, New York (1996)
Roman’kov, V.A.: Automorphisms of groups. Acta Appl. Math. 29(3), 241–280 (1992)
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Communicated by Salvatore Rionero.
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Celentani, M.R., Leone, A. & Nicotera, C. Infinite locally dihedral groups as automorphism groups. Ricerche mat. 63 (Suppl 1), 69–73 (2014). https://doi.org/10.1007/s11587-014-0201-0
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DOI: https://doi.org/10.1007/s11587-014-0201-0