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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Finite groups with prime $p$ 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: 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 ${C_G}(X)$ 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: $|{N_G}(X):{C_G}(X)| = 2$, we have proved that one of the following holds for these groups, hereafter designated as G: (A) G is isomorphic to ${L_2}(q)$, where $q = 2ps \pm 1$; (B) there exists a normal subgroup ${G_0}$ of odd index in G, and a normal subgroup N of ${G_0}$ of index 2 such that $G = N\langle \sigma \rangle$ where ${C_G}(X) = X \times \langle \sigma \rangle$.


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Keywords: Trivial intersection set, <I>p</I>-blocks of defect 1, irreducible characters of defect 0
Article copyright: © Copyright 1976 American Mathematical Society