Algebra & Number Theory

A 2-block splitting in alternating groups

Christine Bessenrodt

Full-text: Open access


In 1956, Brauer showed that there is a partitioning of the p-regular conjugacy classes of a group according to the p-blocks of its irreducible characters with close connections to the block theoretical invariants. In a previous paper, the first explicit block splitting of regular classes for a family of groups was given for the 2-regular classes of the symmetric groups. Based on this work, the corresponding splitting problem is investigated here for the 2-regular classes of the alternating groups. As an application, an easy combinatorial formula for the elementary divisors of the Cartan matrix of the alternating groups at p=2 is deduced.

Article information

Algebra Number Theory, Volume 3, Number 7 (2009), 835-846.

Received: 9 December 2008
Revised: 4 August 2009
Accepted: 5 August 2009
First available in Project Euclid: 20 December 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 20C15: Ordinary representations and characters
Secondary: 20C20: Modular representations and characters 20C30: Representations of finite symmetric groups

alternating groups $p$-regular conjugacy classes irreducible characters Brauer characters $p$-blocks Cartan matrix


Bessenrodt, Christine. A 2-block splitting in alternating groups. Algebra Number Theory 3 (2009), no. 7, 835--846. doi:10.2140/ant.2009.3.835.

Export citation


  • C. Bessenrodt, “The 2-block splitting in symmetric groups”, Algebra Number Theory 1:2 (2007), 223–238.
  • C. Bessenrodt and J. B. Olsson, “The $2$-blocks of the covering groups of the symmetric groups”, Adv. Math. 129:2 (1997), 261–300.
  • C. Bessenrodt and J. B. Olsson, “Spin representations and powers of 2”, Algebr. Represent. Theory 3:3 (2000), 289–300.
  • C. Bessenrodt and J. B. Olsson, “On character tables related to the alternating groups”, Sém. Lothar. Combin. 52 (2004), Art. B52c.
  • R. Brauer, “Zur Darstellungstheorie der Gruppen endlicher Ordnung”, Math. Z. 63 (1956), 406–444.
  • J. W. L. Glaisher, “A theorem in partitions”, Messenger of Math. 12 (1883), 158–170.
  • G. James and A. Kerber, The representation theory of the symmetric group, Encyclopedia of Mathematics and its Applications 16, Addison-Wesley Publishing Co., Reading, MA, 1981.
  • J. B. Olsson, “Lower defect groups in symmetric groups”, J. Algebra 104:1 (1986), 37–56.
  • J. B. Olsson, Combinatorics and representations of finite groups, Vorlesungen aus dem Fachbereich Mathematik 20, Universität Essen, 1993.
  • J. B. Olsson, “Regular character tables of symmetric groups”, Electron. J. Combin. 10 (2003), Note 3.
  • K. Uno and H.-F. Yamada, “Elementary divisors of Cartan matrices for symmetric groups”, J. Math. Soc. Japan 58:4 (2006), 1031–1036.