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On groups with extremal blocks

Published online by Cambridge University Press:  17 April 2009

Marcel Herzog
Marcel Herzog
Affiliation:
Department of Mathematics, Institute of Advanced Studies, Australian National University, Canberra, ACT.
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Abstract

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Let G be a finite group. It is shown that G is 2-closed if and only if

(a) every 2-block of G has full defect, and

(b) every Sylow 2-intersection is centralized by a Sylow 2-subgroup of G.

As a consequence it is shown that G is a TI-group if and only if every 2-block of G has either full defect or defect zero and (b) holds. This result and a theorem of Kwok yield complete characterizations of finite groups with certain relations being satisfied by every nonprincipal irreducible character.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 14 , Issue 3 , June 1976 , pp. 325 - 330
Copyright
Copyright © Australian Mathematical Society 1976

References

[1]Brauer, Richard, “On blocks and sections in finite groups. I”, Amer. J. Math. 89 (1967), 1115–1136.CrossRefGoogle Scholar
[2]Feit, Walter, Characters of finite groups (Benjamin, New York, Amsterdam, 1967).Google Scholar
[3]Feit, Walter, Representation of finite groups (Department of Mathematics, Yale University, New Haven, Connecticut, 1969).Google Scholar
[4]Fong, P., “On the characters of p-solvable groups”, Trans. Amer. Math. Soc. 98 (1961), 263–284.Google Scholar
[5]Gomi, Kensaku, “Finite groups with central Sylow 2-intersections”, J. Math. Soc. Japan 25 (1973), 342–355.Google Scholar
[6]Harada, Koichiro, “On groups all of whose 2-blocks have the highest defects”, Nagoya Math. J. 32 (1968), 283–286.CrossRefGoogle Scholar
[7]Kwok, Chung-Mo, “A characterization of PSL(2, 2m)”, J. Algebra 34 (1975), 288–291.CrossRefGoogle Scholar
[8]Suzuki, Michio, “Finite groups of even order in which Sylow 2-groups are independent”, Ann. of Math. (2) 80 (1964), 58–77.CrossRefGoogle Scholar
[9]Thompson, John G., “Defect groups are Sylow intersections”, Math. Z. 100 (1967), 146.CrossRefGoogle Scholar