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Finite Groups with Abelian Automorphism Groups: A Survey

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Group Theory and Computation

Part of the book series: Indian Statistical Institute Series ((INSIS))

Abstract

In this article, we present an extensive survey on the developments in the theory of non-abelian finite groups with abelian automorphism groups, and pose some problems and further research directions.

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References

  1. J.E. Adney, T. Yen, Automorphisms of \(p\)-group. Ill. J. Math. 9, 137–143 (1965)

    Google Scholar 

  2. G. Ban, S. Yu, Minimal abelian groups that are not automorphism groups. Arch. Math. 70, 427–434 (1998)

    Article  MathSciNet  Google Scholar 

  3. Y. Berkovich, Z. Janko, Groups of Prime Power Order, vol. 2 (Walter de Gruyter, New York, 2008)

    MATH  Google Scholar 

  4. J.N.S. Bidwell, M.J. Curran, D.J. McCaughan, Automorphisms of direct products of finite groups. Arch. Math. 86, 481–489 (2006)

    Article  MathSciNet  Google Scholar 

  5. A. Caranti, A Module theoretic approach to abelian automorphism groups. Isr. J. Math. 205, 235–246 (2015)

    Article  MathSciNet  Google Scholar 

  6. A. Caranti, Erratum to module-theoretic approach to abelian automorphism groups. Isr. J. Math. 215, 1025–1026 (2016)

    Article  Google Scholar 

  7. M.J. Curran, Semi-direct product groups with abelian automorphism groups. J. Austral. Math. Soc. (Ser. A) 42, 84–91 (1987)

    Google Scholar 

  8. M.J. Curran, Direct products with abelian automorphism groups. Commun. Algebra 35, 389–397 (2007)

    Article  MathSciNet  Google Scholar 

  9. R. Dark, C.M. Scoppola, On Camina groups of prime power order. J. Algebra 181, 787–802 (1996)

    Article  MathSciNet  Google Scholar 

  10. B. E. Earnley, On finite groups whose group of automorphisms is abelian, Ph.D. Thesis, Wayne State University, 1975

    Google Scholar 

  11. R. Faudree, A note on the automorphism group of a p-group. Proc. Am. Math. Soc. 19, 1379–1382 (1968)

    MathSciNet  MATH  Google Scholar 

  12. R. Faudree, Groups in which each element commutes with its epimorphic images. Proc. Am. Math. Soc. 27(2), 236–240 (1971)

    Article  Google Scholar 

  13. S.P. Glasby, \(2\)-groups with every automorphisms central. J. Austral. Math. Soc. (Ser. A) 41, 233–236 (1986)

    Google Scholar 

  14. D. Gorenstein, Finite Groups, 2nd edn. (AMS Chelsea Publication, 1980)

    Google Scholar 

  15. H. Heineken, M. Liebeck, The occurrence of finite groups in the automorphism group of nilpotent groups of class \(2\). Arch. Math. 25, 8–16 (1974)

    Google Scholar 

  16. H. Heineken, Nilpotente Gruppen, deren sm̈atliche Normalteiler charakteristisch sind, Arch. Math. 33 (1979/80), 497 - 503

    Google Scholar 

  17. P.V. Hegarty, Minimal abelian automorphism groups of finite groups. Rend. Sem. Mat. Univ. Padova 94, 121–135 (1995)

    MathSciNet  MATH  Google Scholar 

  18. G. T. Helleloid, A survey on automorphism groups of finite groups, arXiv:math/0610294v2 [math.GR], 25 Oct. 2006

  19. H. Hilton, An Introduction to the Theory of Groups of Finite Order (Oxford at the Clarendon Press, 1908)

    Google Scholar 

  20. C. Hopkins, Non-abelian groups whose groups of isomorphisms are abelian. Ann. Math. 29(1), 508–520 (1927)

    Article  MathSciNet  Google Scholar 

  21. A. Hughes, Automorphisms of nilpotent groups and supersolvable groups. Proc. Symp. Pure Math. 37, 205–207 (1980). AMS

    Google Scholar 

  22. M.H. Jafari, Elementary abelian \(p\)-groups as central automorphism groups. Commun. Algebra 34, 601–607 (2006)

    Google Scholar 

  23. A. Jamali, Some new non-abelian \(2\)-groups with abelian automorphism groups. J. Group Theory 5, 53–57 (2002)

    Google Scholar 

  24. V.K. Jain, M.K. Yadav, On finite \(p\)-groups whose automorphisms are all central. Isr. J. Math. 189, 225–236 (2012)

    Google Scholar 

  25. V.K. Jain, P.K. Rai, M.K. Yadav, On finite \(p\)-groups with abelian automorphism groups. Int. J. Algebra Comput. 23, 1063–1077 (2013)

    Google Scholar 

  26. D. Jonah, M. Konvisser, Some non-abelian \(p\)-groups with abelian automorphism groups. Arch. Math. 26, 131–133 (1975)

    Google Scholar 

  27. T. Karimi, Z.K. Farimani, \(p\)-groups with elementary abelian central automorphism groups. World Appl. Program. 1, 352–354 (2011)

    Google Scholar 

  28. R.D. Kitture, M.K. Yadav, Note on Caranti’s method of construction of Miller groups. Monatsh. Math. 185, 87–101 (2018)

    Google Scholar 

  29. I. Malinowska, p-automorphisms of finite p-groups - problems and questions (Rome, Advances in Group Theory, Aracne Edritice, 2002), pp. 111–127

    Google Scholar 

  30. A. Mahalanobis, Diffe-Hellman key exchange protocol and non-Abelian nilpotent groups. Isr. J. Math. 165, 161–187 (2008)

    Article  MathSciNet  Google Scholar 

  31. I.D. Macdonald, The Theory of Groups (Oxford at the Clarendon Press, 1968)

    Google Scholar 

  32. A. Mann, Some questions about \(p\)-groups. J. Austral. Math. Soc. (Ser. A) 67, 356–379 (1999)

    Google Scholar 

  33. G.A. Miller, A non-abelian group whose group of isomorphisms is abelian. Messenger Math. XVIII, 124–125 (1913-1914)

    Google Scholar 

  34. M. Morigi, On the minimal number of generators of finite non-abelian \(p\)-groups having an abelian automorphism group. Commun. Algebra 23, 2045–2065 (1995)

    Google Scholar 

  35. M. Morigi, On \(p\)-groups with abelian automorphism group. Rend. Sem. Mat. Univ. Padova 92, 47–58 (1994)

    Google Scholar 

  36. R.R. Struik, Some non-abelian \(2\)-groups with abelian automorphism groups. Arch. Math. 39, 299–302 (1982)

    Google Scholar 

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Correspondence to Manoj K. Yadav .

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Kitture, R.D., Yadav, M.K. (2018). Finite Groups with Abelian Automorphism Groups: A Survey. In: Sastry, N., Yadav, M. (eds) Group Theory and Computation. Indian Statistical Institute Series. Springer, Singapore. https://doi.org/10.1007/978-981-13-2047-7_7

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