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 | 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 ${C_G}(X)$ of *X*, where *X* is a subgroup of order 3, is a cyclic trivial intersection set of even order 3*s*. 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 ${C_G}(X) = X \times \langle \sigma \rangle$.

- Richard Brauer,
*On groups whose order contains a prime number to the first power. I*, Amer. J. Math.**64**(1942), 401–420. MR**6537**, DOI https://doi.org/10.2307/2371693 - R. 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**104734** - Richard Brauer and K. A. Fowler,
*On groups of even order*, Ann. of Math. (2)**62**(1955), 565–583. MR**74414**, DOI https://doi.org/10.2307/1970080 - Larry Dornhoff,
*Group representation theory. Part A: Ordinary representation theory*, Marcel Dekker, Inc., New York, 1971. Pure and Applied Mathematics, 7. MR**0347959** - Daniel Gorenstein,
*Finite groups*, Harper & Row, Publishers, New York-London, 1968. MR**0231903**
Donald G. Higman, - 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 York-London, 1962. MR**0144979** - Leo J. Alex,
*Simple groups of order $2^{a}3^{b}5^{c}7^{d}p$*, Trans. Amer. Math. Soc.**173**(1972), 389–399. MR**318291**, DOI https://doi.org/10.1090/S0002-9947-1972-0318291-1 - Leo J. Alex,
*On simple groups of order $2^{a}\cdot 3^{b}\cdot 7^{c}\cdot p$*, J. Algebra**25**(1973), 113–124. MR**320134**, DOI https://doi.org/10.1016/0021-8693%2873%2990079-3 - Leo J. Alex,
*Index two simple groups*, J. Algebra**31**(1974), 262–275. MR**354843**, DOI https://doi.org/10.1016/0021-8693%2874%2990068-4

*Finite groups with*3

*to the first power*, Univ. of Michigan and Univ. of British Columbia (preprint).

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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