Hostname: page-component-84b7d79bbc-x5cpj Total loading time: 0 Render date: 2024-07-29T03:24:54.021Z Has data issue: false hasContentIssue false
  • English
  • Français

Character Theory of Finite Groups with Trivial Intersection Subsets

Published online by Cambridge University Press:  22 January 2016

John H. Walter
John H. Walter*
Affiliation:
University of Illinois, Urbana, Illinois
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.

This paper arose out of an effort to present a result which simplifies the application of the theory of blocks to what is often called the theory of exceptional characters. In order to obtain the most effective use of this result, it is necessary to reformulate part of this theory. Thus we present a development which explains an application of R. Brauer’s main theorem on generalized decomposition numbers. In particular, we improve a result of D. Gorenstein and the author [8; Proposition 25]. These results are needed in a forthcoming paper and will simplify somewhat the use of this theory in existing papers. We are interested principally in determining the values of certain irreducible characters on trivial intersection subsets. The organization of the theory presented here is influenced by an exposition of M. Suzuki [13] and uses concepts introduced by W. Feit and J. G. Thompson [7]. Also it is hoped that this exposition will serve as an introduction to the theory.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 27 , Issue 2 , July 1966 , pp. 515 - 524
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1966

References

[1] Brauer, R., On the connection between the ordinary and modular characters of groups of finite order. Ann. of Math 42 (1941), 926–935.CrossRefGoogle Scholar
[2] Brauer, R., Zur Darstellungstheorie der Gruppen endlicher Ordnung II, Math. Zeit. 72 (1959).CrossRefGoogle Scholar
[3] Brauer, R., Investigations of groups of even order. Proc. Nat. Ac. Sci., 47 (1961), 1891–1893.CrossRefGoogle ScholarPubMed
[4] Brauer, R., Some applications of the theory of blocks of finite groups, I. J. Algebra, 1 (1964), 152–167.CrossRefGoogle Scholar
[5] Dade, E., Lifting group characters, Ann. of Math. 79 (1964), 590–596.CrossRefGoogle Scholar
[6] Feit, W., On a class of doubly transitive permutation groups. 111. J. Math. 4 (1960), 170–186.Google Scholar
[7] Feit, W. and Thompson, J. G., Solvability of groups of odd order. Pac. J. Math. 13 (1963), 775–1029.Google Scholar
[8] Gorenstein, D. and Walter, J. H., On finite groups with dihedral Sylow 2-subgroups, 111. J. Math. 6 (1952), 553–593.Google Scholar
[9] Gorenstein, D. and Walter, J. H., The characterization of finite groups with dihedral Sylow 2-subgroups, I. J. Algebra, 2 (1965), 85–151.CrossRefGoogle Scholar
[10] Iizuka, K., On Brauers theorem on sections in the theory of group characters, Math. Zeit. 75 (1961), 299–304.CrossRefGoogle Scholar
[11] Nagao, H., A proof of Brauers theorem on generalized decomposition numbers, Nagoya Math. J. 22 (1963), 73–79.CrossRefGoogle Scholar
[12] Suzuki, M., On finite groups with cyclic Sylow groups all add primes, Am. J. Math. 77 (1955), 657–691.CrossRefGoogle Scholar
[13] Suzuki, M., Applications of group characters. Proc. Symp. Pur Math. vol. I (Finite Groups), Providence, 1959, 88–99.CrossRefGoogle Scholar