Finite groups with prime to the first power
Author:
Zon I Chang
Journal:
Trans. Amer. Math. Soc. 222 (1976), 267-288
MSC:
Primary 20D25
DOI:
https://doi.org/10.1090/S0002-9947-1976-0427464-2
MathSciNet review:
0427464
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Abstract | References | Similar Articles | Additional Information
Abstract: Earlier D. G. Higman classified the finite groups of order n, such that n is divisible by 3 to the first power, with the assumption that the centralizer of X, where X is a subgroup of order 3, is a cyclic trivial intersection set of even order 3s. In this paper the theorem is generalized to include all prime numbers greater than 3. With an additional assumption:
, we have proved that one of the following holds for these groups, hereafter designated as G:
(A) G is isomorphic to , where
;
(B) there exists a normal subgroup of odd index in G, and a normal subgroup N of
of index 2 such that
where
.
- [1] Richard Brauer, On groups whose order contains a prime number to the first power. I, Amer. J. Math. 64 (1942), 401-420. MR 4, 1. MR 0006537 (4:1e)
- [2] Richard Brauer, Michio Suzuki and G. E. Wall, A characterization of the one-dimensional unimodular projective groups over finite fields, Illinois J. Math. 2 (1958), 718-745. MR 21 #3487. MR 0104734 (21:3487)
- [3] Richard Brauer and K. A. Fowler, On groups of even order, Ann. of Math. (2) 62 (1955), 565-583. MR 17, 580. MR 0074414 (17:580e)
- [4] Larry Dornhoff, Group representation theory. Part A: Ordinary representation theory; Part B: Modular representation theory, Pure and Appl. Math., vol. 7, Dekker, New York, 1972. MR 50 #458a, b. MR 0347959 (50:458a)
- [5] Daniel Gorenstein, Finite groups, Harper and Row, New York and London, 1968. MR 38 #229. MR 0231903 (38:229)
- [6] Donald G. Higman, Finite groups with 3 to the first power, Univ. of Michigan and Univ. of British Columbia (preprint).
- [7] Charles W. Curtis and Irving Reiner, Representation theory of finite groups and associative algebras, Pure and Appl. Math., vol. 11, Interscience, New York, 1962. MR 26 #2519. MR 0144979 (26:2519)
- [8]
Leo J. Alex, Simple groups of order
, Trans. Amer. Math. Soc. 173 (1972), 389-399. MR 47 #6838. MR 0318291 (47:6838)
- [9]
-, On simple groups of order
, J. Algebra 25 (1973), 113-124. MR 47 #8675. MR 0320134 (47:8675)
- [10] -, Index two simple groups, J. Algebra 31 (1974), 262-275. MR 50 #7320. MR 0354843 (50:7320)
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Additional Information
DOI:
https://doi.org/10.1090/S0002-9947-1976-0427464-2
Keywords:
Trivial intersection set,
p-blocks of defect 1,
irreducible characters of defect 0
Article copyright:
© Copyright 1976
American Mathematical Society