Finite groups with prime to the first power
Author:
Zon I Chang
Journal:
Trans. Amer. Math. Soc. 222 (1976), 267288
MSC:
Primary 20D25
MathSciNet review:
0427464
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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 0006537
(4,1e)
 [2]
R.
Brauer, Michio
Suzuki, and G.
E. Wall, A characterization of the onedimensional unimodular
projective groups over finite fields, Illinois J. Math.
2 (1958), 718–745. 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 0074414
(17,580e)
 [4]
Larry
Dornhoff, Group representation theory. Part A: Ordinary
representation theory, Marcel Dekker Inc., New York, 1971. Pure and
Applied Mathematics, 7. MR 0347959
(50 #458a)
 [5]
Daniel
Gorenstein, Finite groups, Harper & Row Publishers, New
York, 1968. 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 Applied Mathematics, Vol. XI, Interscience
Publishers, a division of John Wiley & Sons, New YorkLondon, 1962. MR 0144979
(26 #2519)
 [8]
Leo
J. Alex, Simple groups of order
2^{𝑎}3^{𝑏}5^{𝑐}7^{𝑑}𝑝, Trans. Amer. Mat. Soc. 173 (1972), 389–399. MR 0318291
(47 #6838), http://dx.doi.org/10.1090/S00029947197203182911
 [9]
Leo
J. Alex, On simple groups of order
2^{𝑎}⋅3^{𝑏}⋅7^{𝑐}⋅𝑝, J.
Algebra 25 (1973), 113–124. MR 0320134
(47 #8675)
 [10]
Leo
J. Alex, Index two simple groups, J. Algebra
31 (1974), 262–275. MR 0354843
(50 #7320)
 [1]
 Richard Brauer, On groups whose order contains a prime number to the first power. I, Amer. J. Math. 64 (1942), 401420. MR 4, 1. MR 0006537 (4:1e)
 [2]
 Richard Brauer, Michio Suzuki and G. E. Wall, A characterization of the onedimensional unimodular projective groups over finite fields, Illinois J. Math. 2 (1958), 718745. 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), 565583. 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), 389399. MR 47 #6838. MR 0318291 (47:6838)
 [9]
 , On simple groups of order , J. Algebra 25 (1973), 113124. MR 47 #8675. MR 0320134 (47:8675)
 [10]
 , Index two simple groups, J. Algebra 31 (1974), 262275. MR 50 #7320. MR 0354843 (50:7320)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947197604274642
PII:
S 00029947(1976)04274642
Keywords:
Trivial intersection set,
pblocks of defect 1,
irreducible characters of defect 0
Article copyright:
© Copyright 1976 American Mathematical Society
