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Finite Linear Groups of Degree Seven. I

Published online by Cambridge University Press:  20 November 2018

David B. Wales
David B. Wales*
Affiliation:
California Institute of Technology, Pasadena, California
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1. 1. This paper is the second in a series of papers discussing linear groups of prime degree, the first being (8). In this paper we discuss only linear groups of degree 7. Thus, G is a finite group with a faithful irreducible complex representation Xof degree 7 which is unimodular and primitive. The character of Xis x- The notation of (8) is used except here p= 7. Thus Pis a 7-Sylow group of G.In §§ 2 and 3 some general theorems about the 3-Sylow group and 5-Sylow group are given. In § 4 the statement of the results when Ghas a non-abelian 7-Sylow group is given. This corresponds to the case |P| =73 or |P|= 74. The proof is given in §§ 5 and 6. In a subsequent paper the results when Pis abelian will be given.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 21 , 1969 , pp. 1042 - 1053
Copyright
Copyright © Canadian Mathematical Society 1969

References

1. Blichfeldt, H. F., Finite collineation groups (University of Chicago Press, Chicago, Illinois, 1917).Google Scholar
2. Brauer, R., On groups whose order contains a prime number to the first power. I ; II, Amer. J. Math. 64 (1942), 401-420; 64 (1942), 421–440 Google Scholar
3. Brauer, R., Uber endliche lineare Gruppen von Primzahlgrad, Math. Ann. 169 (1967), 73–96.Google Scholar
4. Brauer, R. and Leonard, H. S., Jr., On finite groups with an abelian Sylow group, Can. J. Math. 14 (1962), 436–450.Google Scholar
5. Clifford, A. H., Representations induced in an invariant subgroup, Ann. of Math. 88 (1937), 533–550.Google Scholar
6. Hayden, S., On finite linear groups whose order contains a prime larger than the degree, Thesis, Harvard University, Cambridge, Massachusetts, 1963.Google Scholar
7. Schur, I., Über eine Klasse von endlichen Gruppen linearer Substitutionen, Sitzber. Preuss. Akad. Wiss. Berlin 1905, 77–91.Google Scholar
7. Wales, D. B., Finite linear groups of prime degree, Can. J. Math. 21 (1969), 1025–1041.Google Scholar