Reflection Groups: "These
are groups (acting on a finite dimensional vector space) generated
by reflections: elements that fix a hyperplane pointwise. They include
the Weyl and Coxeter groups, complex reflection groups... and
reflection groups over arbitrary fields." -- Anne V. Shepler, home page, 2007
Note that this definition includes the simple group of order 168, which
is isomorphic to GL(3,2), the general linear group of three
dimensions over the two-element field.
We may visualize this group as generated by reflections in the form of
certain permutations of parts of the eightfold cube (below) that
all fix the same (arbitrarily selected) subcube.
There are larger easily-visualized finite
reflection groups over the binary field (and, of course,
a smaller such group). See Finite
Geometry of the Square and Cube.
For reflection groups over other finite fields, see the following:
FINITE LINEAR GROUPS GENERATED BY REFLECTIONS
by A. E. Zalesskii and V. N. Serezhkin
Abstract
A
complete classification is given of all finite irreducible linear
groups generated by reflections over an arbitrary field of
characteristic not 2.
Bibliography: 10 titles.
DOI
10.1070/IM1981v017n03ABEH001369
Citation
A E
Zalesskii,
V N Serezhkin, "FINITE LINEAR GROUPS GENERATED BY
REFLECTIONS", MATH
USSR IZV, 1981, 17 (3), 477-503.
