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Transactions of the American Mathematical Society
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
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: $ \vert{N_G}(X):{C_G}(X)\vert = 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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DOI: http://dx.doi.org/10.1090/S0002-9947-1976-0427464-2
PII: S 0002-9947(1976)0427464-2
Keywords: Trivial intersection set, p-blocks of defect 1, irreducible characters of defect 0
Article copyright: © Copyright 1976 American Mathematical Society