Hostname: page-component-78c5997874-m6dg7 Total loading time: 0 Render date: 2024-11-13T02:40:58.649Z Has data issue: false hasContentIssue false
  • English
  • Français

On the Block of Defect Zero

Published online by Cambridge University Press:  22 January 2016

Yukio Tsushima
Yukio Tsushima*
Affiliation:
Osaka City University
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let G be a finite group and let p be a fixed prime number. If D is any p-subgroup of G, then the problem whether there exists a p-block with D as its defect group is reduced to whether NG(D)/D possesses a p-block of defect 0. Some necessary or sufficient conditions for a finite group to possess a p-block of defect 0 have been known (Brauer-Fowler [1], Green [3], Ito [4] [5]). In this paper we shall show that the existences of such blocks depend on the multiplicative structures of the p-elements of G. Namely, let p be a prime divisor of p in an algebraic number field which is a splitting one for G, o the ring of p-integers and k = o/p, the residue class field.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 44 , December 1971 , pp. 57 - 59
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1971

References

[1] Brauer, R. and Fowler, K.A., On groups of even order, Ann. of Math. 62 (1955), 565–583.CrossRefGoogle Scholar
[2] Curtis, C.W. and Reiner, I., Representation theory of finite groups and associative algebras, Wiley, New York, 1962.Google Scholar
[3] Green, J.A., Blocks of modular representation, Math. Zeitshr. 79 (1962) 100–115.CrossRefGoogle Scholar
[4] Itô, N., On the characters of soluble groups, Nagoya Math. J. 3 (1951) 31–48.Google Scholar
[5] Itô, N., Note on the characters of solvable groups, ibid, 39 (1970) 23–28.Google Scholar
[6] Tsushima, Y., On the annihilator ideal of the radical of a group algebra, Osaka J. (to appear)Google Scholar