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On the Block of Defect Zero

Published online by Cambridge University Press:  22 January 2016

Yukio Tsushima
Yukio Tsushima*
Affiliation:
Osaka City University
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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

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[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