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Verification of Dade's Conjecture for Janko GroupJ3

✍ Scribed by Sonja Kotlica

Publisher: Elsevier Science
Year: 1997
Tongue: English
Weight: 400 KB
Volume: 187
Category: Article
ISSN: 0021-8693
DOI: 10.1006/jabr.1996.6836

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✦ Synopsis

In 3 Dade made a conjecture expressing the number k B, d of characters of a given defect d in a given p-block B of a finite group G in terms of the Ž . corresponding numbers k b, d for blocks b of certain p-local subgroups of G. w x Several different forms of this conjecture are given in 5 .

Dade claims that the most complicated form of this conjecture, called the w x ''Inductive Conjecture 5.8'' in 5 , will hold for all finite groups if it holds for all covering groups of finite simple groups. In this paper we verify the inductive Ž conjecture for all covering groups of the third Janko group J in the notation of 3 w x. the Atlas 1 . This is one step in the inductive proof of the conjecture for all finite groups.

Certain properties of J simplify our task. The Schur Multiplier of J is cyclic of 3 3 Ž w x . order 3 see 1, p. 82 . Hence, there are just two covering groups of J , namely J 3 3

itself and a central extension 3 и J of J by a cyclic group Z of order 3. We treat 3 3

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