Hall-Higman type theorems. II

Berger, T. R.

doi:10.1090/S0002-9947-1975-0399229-0

Hall-Higman type theorems. II
HTML articles powered by AMS MathViewer

by T. R. Berger

Trans. Amer. Math. Soc. 205 (1975), 47-69

DOI: https://doi.org/10.1090/S0002-9947-1975-0399229-0

PDF | Request permission

Abstract:

This paper continues the investigations of this series. Suppose ${\mathbf {K}} = \text {GF} {\text {(}}r{\text {)}}$ is a field for a prime $r;G$-is a nilpotent; $V$ is a nonsingular symplectic space with form $g$; and $V$ is a faithful irreducible ${\mathbf {K}}[G]$-module where $G$ fixes the form $g$. This paper describes completely the structure of $G$ and its representation upon $V$ when $G$ is symplectic primitive. This latter condition is described in §4 and is a primitivity condition.

References

T. R. Berger, Automorphisms of solvable groups, J. Algebra 27 (1973), 311–340. MR 347978, DOI 10.1016/0021-8693(73)90108-7
T. R. Berger, Hall-Higman type theorems. I, Canadian J. Math. 26 (1974), 513–531. MR 399228, DOI 10.4153/CJM-1974-048-5
T. R. Berger, Hall-Higman type theorems. III, Trans. Amer. Math. Soc. 228 (1977), 47–83. MR 437627, DOI 10.1090/S0002-9947-1977-0437627-9
T. R. Berger, Hall-Higman type theorems. V, Pacific J. Math. 73 (1977), no. 1, 1–62. MR 482784

Hall-Higman type theorems

Thomas R. Berger, Nilpotent fixed point free automorphism groups of solvable groups, Math. Z. 131 (1973), 305–312. MR 338174, DOI 10.1007/BF01174905
T. R. Berger, Characters and derived length in groups of odd order, J. Algebra 39 (1976), no. 1, 199–207. MR 396729, DOI 10.1016/0021-8693(76)90070-3
T. R. Berger, Primitive solvable groups, J. Algebra 33 (1975), 9–21. MR 360806, DOI 10.1016/0021-8693(75)90128-3

-length and the derived length of a Sylow $2$-subgroup

P. Hall and Graham Higman, On the $p$-length of $p$-soluble groups and reduction theorems for Burnside’s problem, Proc. London Math. Soc. (3) 6 (1956), 1–42. MR 72872, DOI 10.1112/plms/s3-6.1.1