Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Journal of Group Theory

Editor-in-Chief: Parker, Christopher W.

Managing Editor: Wilson, John S. / Khukhro, Evgenii I. / Kramer, Linus


IMPACT FACTOR 2018: 0.470
5-year IMPACT FACTOR: 0.520

CiteScore 2018: 0.53

SCImago Journal Rank (SJR) 2018: 0.566
Source Normalized Impact per Paper (SNIP) 2018: 1.047

Mathematical Citation Quotient (MCQ) 2018: 0.48

Online
ISSN
1435-4446
See all formats and pricing
More options …
Volume 20, Issue 2

Issues

Finite p-groups of maximal class with ‘large’ automorphism groups

Heiko Dietrich / Bettina Eick
Published Online: 2016-09-14 | DOI: https://doi.org/10.1515/jgth-2016-0036

Abstract

The classification of p-groups of maximal class still is a wide open problem. Coclass Conjecture W proposes a way to approach such a classification: It suggests that the coclass graph 𝒢 associated with the p-groups of maximal class can be determined from a finite subgraph using certain periodic patterns. Here we consider the subgraph 𝒢 of 𝒢 associated with those p-groups of maximal class whose automorphism group orders are divisible by p-1. We describe the broad structure of 𝒢 by determining its so-called skeleton. We investigate the smallest interesting case p=7 in more detail using computational tools, and propose an explicit version of Conjecture W for 𝒢 for arbitrary p7. Our results are the first explicit evidence in support of Conjecture W for a coclass graph of infinite width.

References

  • [1]

    Blackburn N., On a special class of p-groups, Acta Math. 100 (1958), 45–92. Google Scholar

  • [2]

    Couson M., Character degrees of finite p-groups by coclass, J. Algebra 418 (2014), 91–109. Google Scholar

  • [3]

    Dietrich H., A new periodic pattern in the graph of p-groups of maximal class, Bull. Lond. Math. Soc. 42 (2010), 1073–1088. Google Scholar

  • [4]

    Dietrich H., Periodic patterns in the graph of p-groups of maximal class, J. Group Theory 13 (2010), 851–871. Google Scholar

  • [5]

    Dietrich H., Eick B. and Feichtenschlager D., Investigating p-groups by coclass with GAP, Computational Group Theory and the Theory of Groups (Davidson 2007), Contemp. Math. 470, American Mathematical Society, Providence (2008), 45–61. Google Scholar

  • [6]

    Dixon J. D., du Sautoy M. P. F., Mann A. and Segal D., Analytic Pro-p-groups, 2nd ed., Cambridge University Press, Cambridge, 2003. Google Scholar

  • [7]

    Eick B., Metabelian p-groups and coclass theory, J. Algebra 421 (2015), 102–118. Google Scholar

  • [8]

    Eick B. and Leedham-Green C. R., On the classification of prime-power groups by coclass, Bull. Lond. Math. Soc. 40 (2008), 274–288. Google Scholar

  • [9]

    Eick B., Leedham-Green C. R., Newman M. F. and O’Brien E. A., On the classification of groups of prime-power order by coclass: The 3-groups of coclass 2, Internat. J. Algebra Comput. 23 (2013), 1243–1288. Google Scholar

  • [10]

    Leedham-Green C. R. and McKay S., On the classification of p-groups of maximal class, Quart. J. Math. Oxford 35 (1984), 293–304. Google Scholar

  • [11]

    Leedham-Green C. R. and McKay S., The Structure of Groups of Prime Power Order, London Math. Soc. Monogr. Ser. (N.S.) 27, Oxford University Press, Oxford, 2002. Google Scholar

  • [12]

    Leedham-Green C. R. and Newman M. F., Space groups and groups of prime-power order I, Arch. Math. 35 (1980), 193–203. Google Scholar

  • [13]

    Neukirch J., Algebraische Zahlentheorie, Springer, Berlin, 1992. Google Scholar

  • [14]

    Newman M. F., Groups of prime-power order, Groups (Canberra 1989), Lecture Notes in Math. 1456, Springer, Berlin (1990), 49–62. Google Scholar

  • [15]

    Newman M. F. and O’Brien E. A., Classifying 2-groups using coclass, Trans. Amer. Math. Soc. 351 (1990), 131–169. Google Scholar

  • [16]

    O’Brien E. A., The p-group generation algorithm, J. Symbolic Comput. 9 (1990), 677–698. Google Scholar

  • [17]

    Robinson D. J. S., Applications of cohomology groups to the theory of groups, Groups (St. Andrews 1981), London Math. Soc. Lecture Note Ser. 71, Cambridge University Press, Cambridge (1981), 46–80. Google Scholar

  • [18]

    du Sautoy M., Counting p-groups and nilpotent groups, Publ. Math. Inst. Hautes Etudes Sci. 92 (2001), 63–112. Google Scholar

  • [19]

    Wiman A., Über p-Gruppen mit maximaler Klasse, Acta Math. 88 (1952), 317–346. Google Scholar

  • [20]

    GAP – Groups, Algorithms and Programming, http://www.gap-system.org.

About the article


Received: 2016-03-15

Published Online: 2016-09-14

Published in Print: 2017-03-01


This research was supported by a Go8-DAAD Joint Research Co-operation Scheme, project “Groups of Prime-Power Order and Coclass Theory”. The second author was also supported by an ARC DECRA (Australia), project DE140100088.


Citation Information: Journal of Group Theory, Volume 20, Issue 2, Pages 227–256, ISSN (Online) 1435-4446, ISSN (Print) 1433-5883, DOI: https://doi.org/10.1515/jgth-2016-0036.

Export Citation

© 2017 by De Gruyter.Get Permission

Citing Articles

Here you can find all Crossref-listed publications in which this article is cited. If you would like to receive automatic email messages as soon as this article is cited in other publications, simply activate the “Citation Alert” on the top of this page.

[1]
Heiko Dietrich and Subhrajyoti Saha
International Journal of Algebra and Computation, 2018, Page 1

Comments (0)

Please log in or register to comment.
Log in