Skip to main content

Defect Groups of p-Rank 2

  • 890 Accesses

Part of the Lecture Notes in Mathematics book series (LNM,volume 2127)

Abstract

The p-rank of a finite p-group is the maximal rank of an abelian subgroup. For odd primes the p-groups of p-rank at most 2 are classified by Blackburn. We use this classification in order to prove Olsson’s Conjecture for all blocks with defect groups of p-rank at most 2 provided p > 3. We also develop general methods which deal with controlled blocks.

Keywords

  • Defect Group
  • Abelian Subgroup
  • Maximal Rank
  • Elementary Group Theory
  • Principal Block

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

This is a preview of subscription content, access via your institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   39.99
Price excludes VAT (Canada)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   54.99
Price excludes VAT (Canada)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

References

  1. An, J.: Dade’s conjecture for the Tits group. New Zealand J. Math. 25(2), 107–131 (1996)

    MATH  MathSciNet  Google Scholar 

  2. An, J.: The Alperin and Dade conjectures for Ree groups \(^{2}F_{4}(q^{2})\) in non-defining characteristics. J. Algebra 203(1), 30–49 (1998)

    CrossRef  MATH  MathSciNet  Google Scholar 

  3. An, J.: Controlled blocks of the finite quasisimple groups for odd primes. Adv. Math. 227(3), 1165–1194 (2011)

    CrossRef  MATH  MathSciNet  Google Scholar 

  4. An, J., Eaton, C.W.: Blocks with extraspecial defect groups of finite quasisimple groups. J. Algebra 328, 301–321 (2011)

    CrossRef  MATH  MathSciNet  Google Scholar 

  5. An, J., O’Brien, E.A.: The Alperin and Dade conjectures for the O’Nan and Rudvalis simple groups. Commun. Algebra 30(3), 1305–1348 (2002)

    MATH  MathSciNet  Google Scholar 

  6. An, J., O’Brien, E.A., Wilson, R.A.: The Alperin weight conjecture and Dade’s conjecture for the simple group J 4. LMS J. Comput. Math. 6, 119–140 (2003)

    CrossRef  MATH  MathSciNet  Google Scholar 

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

    CrossRef  MATH  MathSciNet  Google Scholar 

  8. Conway, J.H., Curtis, R.T., Norton, S.P., Parker, R.A., Wilson, R.A.: ATLAS of finite groups. Oxford University Press, Eynsham (1985). Maximal subgroups and ordinary characters for simple groups, With computational assistance from J.G. Thackray

    Google Scholar 

  9. Díaz, A., Ruiz, A., Viruel, A.: All p-local finite groups of rank two for odd prime p. Trans. Am. Math. Soc. 359(4), 1725–1764 (2007)

    Google Scholar 

  10. Guralnick, R.M.: Commutators and commutator subgroups. Adv. Math. 45(3), 319–330 (1982)

    CrossRef  MATH  MathSciNet  Google Scholar 

  11. Hendren, S.: Extra special defect groups of order p 3 and exponent p. J. Algebra 313(2), 724–760 (2007)

    Google Scholar 

  12. Héthelyi, L., Külshammer, B., Sambale, B.: A note on olsson’s conjecture. J. Algebra 398, 364–385 (2014)

    CrossRef  MATH  MathSciNet  Google Scholar 

  13. Huppert, B.: Endliche Gruppen. I. Die Grundlehren der Mathematischen Wissenschaften, Band 134. Springer, Berlin (1967)

    Google Scholar 

  14. Isaacs, I.M.: Finite Group Theory. Graduate Studies in Mathematics, vol. 92. American Mathematical Society, Providence (2008)

    Google Scholar 

  15. Kessar, R., Stancu, R.: A reduction theorem for fusion systems of blocks. J. Algebra 319(2), 806–823 (2008)

    CrossRef  MATH  MathSciNet  Google Scholar 

  16. Narasaki, R., Uno, K.: Isometries and extra special Sylow groups of order p 3. J. Algebra 322(6), 2027–2068 (2009)

    CrossRef  MATH  MathSciNet  Google Scholar 

  17. Ruiz, A., Viruel, A.: The classification of p-local finite groups over the extraspecial group of order p 3 and exponent p. Math. Z. 248(1), 45–65 (2004)

    Google Scholar 

  18. Sambale, B.: Further evidence for conjectures in block theory. Algebra Number Theory 7(9), 2241–2273 (2013)

    CrossRef  MATH  MathSciNet  Google Scholar 

  19. Stancu, R.: Control of fusion in fusion systems. J. Algebra Appl. 5(6), 817–837 (2006)

    CrossRef  MATH  MathSciNet  Google Scholar 

  20. Uno, K.: Conjectures on character degrees for the simple Thompson group. Osaka J. Math. 41(1), 11–36 (2004)

    MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and Permissions

Copyright information

© 2014 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Sambale, B. (2014). Defect Groups of p-Rank 2. In: Blocks of Finite Groups and Their Invariants. Lecture Notes in Mathematics, vol 2127. Springer, Cham. https://doi.org/10.1007/978-3-319-12006-5_11

Download citation