Abstract
The non-abelian groups with abelian group of automorphisms are widely studied. Following Earnley, such groups are called Miller groups, since the first example of such a group was given by G.A. Miller in 1913. Many other examples of Miller p-groups have been constructed by several authors. Recently, Caranti (Isr J Math 205: 235–246, 2015) provided module theoretic methods for constructing non-special Miller p-groups from special Miller p-groups. By constructing examples, we show that these methods do not always work. We also provide a sufficient condition on special Miller p-group for which the methods of Caranti work.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bidwell, J.N.S., Curran, M.J., McCaughan, D.J.: Automorphisms of direct products of finite groups. Arch. Math. 86, 481–489 (2006)
Bosma, W., Cannon, J., Playoust, C.: The Magma algebra system. I: the user language. J. Symb. Comput. 24, 235–265 (1997)
Caranti, A.: A Module theoretic approach to abelian automorphism groups. Isr. J. Math. 205, 235–246 (2015)
Curran, M.J., McChaughan, D.J.: Central automorphisms that are almost inner. Commun. Algebra 29, 2081–2087 (2001)
Earnley, B.E.: On finite groups whose group of automorphisms is abelian. Ph.D. Thesis, Waybe State University (1975)
Gorenstein, D.: Finite Groups. Second Ed. AMS Chelsea Pub. (1980)
Hopkins, C.: Non-abelian groups whose groups of isomorphisms are abelian. Ann. Math. 29(1), 508–520 (1927)
Heineken, H., Liebeck, M.: The occurrence of finite groups in the automorphism group of nilpotent groups of class \(2\). Arch. Math. 25, 8–16 (1974)
Hilton, H.: An Introduction to the Theory of Groups of Finite Order. Oxford Clarendon Press, Oxford (1908)
Jain, V.K., Yadav, M.K.: On finite \(p\)-groups whose automorphisms are all central. Isr. J. Math. 189, 225–236 (2012)
Jain, V.K., Rai, P.K., Yadav, M.K.: On finite \(p\)-groups with abelian automorphism groups. Int. J. Algebra Comput. 23, 1063–1077 (2013)
Jonah, D., Konvisser, M.: Some non-abelian \(p\)-groups with abelian automorphism groups. Arch. Math. 26, 131–133 (1975)
Miller, G.A.: A non-abelian group whose group of isomorphisms is abelian. Messenger Math. XVIII, 124–125 (1913–1914)
Morigi, M.: On \(p\)-groups with abelian automorphism group. Rendiconti del Seminario Matematico della Universita di Padova 92, 47–58 (1994)
Morigi, M.: On the minial number of generators of finite non-abelian \(p\)-groups having an abelian automorphism group. Commun. Algebra 23, 2045–2065 (1995)
The GAP Group.: GAP—Groups, Algorithms, and Programming, Version 4.8.3, 2016. http://www.gap-system.org
Zassenhaus, H.: The Theory of Groups. Chelsea Publishing Company, New York (1949)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by J. S. Wilson.
Rights and permissions
About this article
Cite this article
Kitture, R.D., Yadav, M.K. Note on Caranti’s method of construction of Miller groups. Monatsh Math 185, 87–101 (2018). https://doi.org/10.1007/s00605-016-0976-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00605-016-0976-z