Hall-Higman type theorems. II

Author:
T. R. Berger

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

MSC:
Primary 20C05

MathSciNet review:
0399229

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Abstract: This paper continues the investigations of this series. Suppose GF is a field for a prime -is a nilpotent; is a nonsingular symplectic space with form ; and is a faithful irreducible -module where fixes the form . This paper describes completely the structure of and its representation upon when is symplectic primitive. This latter condition is described in §4 and is a primitivity condition.

**[1]**T. R. Berger,*Automorphisms of solvable groups*, J. Algebra**27**(1973), 311–340. MR**0347978****[2]**T. R. Berger,*Hall-Higman type theorems. I*, Canad. J. Math.**26**(1974), 513–531. MR**0399228****[3]**T. R. Berger,*Hall-Higman type theorems. III*, Trans. Amer. Math. Soc.**228**(1977), 47–83. MR**0437627**, 10.1090/S0002-9947-1977-0437627-9**[4]**T. R. Berger,*Hall-Higman type theorems. V*, Pacific J. Math.**73**(1977), no. 1, 1–62. MR**482784****[5]**-,*Hall-Higman type theorems*. VIII, Proc. London Math. Soc. (to appear).**[6]**Thomas R. Berger,*Nilpotent fixed point free automorphism groups of solvable groups*, Math. Z.**131**(1973), 305–312. MR**0338174****[7]**T. R. Berger,*Characters and derived length in groups of odd order*, J. Algebra**39**(1976), no. 1, 199–207. MR**0396729****[8]**T. R. Berger,*Primitive solvable groups*, J. Algebra**33**(1975), 9–21. MR**0360806****[9]**T. R. Berger and F. Gross,*-length and the derived length of a Sylow -subgroup*, Proc. London. Math. Soc. (to appear).**[10]**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**0072872**

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

Keywords:
Hall-Higman Theorem B,
representation theory,
group theory,
minimal module,
symplectic primitive module

Article copyright:
© Copyright 1975
American Mathematical Society