Log24

The non-Coxeter simple reflection group of order 168
is a counterexample to the statement that
"Every finite reflection group is a Coxeter group."

The counterexample is based on a definition of "reflection group"
that includes reflections defined over finite fields.

Today I came across a 1911 paper that discusses the counterexample.
Of course, Coxeter groups were undefined in 1911, but the paper, by
Howard H. Mitchell, discusses the simple order-168 group as a reflection group .

(Naturally, Mitchell's definition of "reflection" and his statement that

"The discussion of the binary groups
applies also to the case p = 2."

should be approached with care.)

A review of this topic might be appropriate for Jessica Fintzen's 2012 fall tutorial at Harvard
on reflection groups and Coxeter groups. The syllabus for the tutorial states that
"finite Coxeter groups correspond precisely to finite reflection groups." This statement
is based on Fintzen's definition of "reflection group"—

"Reflection groups are— as their name indicates—
groups generated by reflections across
hyperplanes of Rⁿ which contain the origin."

For some background, see William Kantor's 1981 paper "Generation of Linear Groups"
(quoted at the finitegeometry.org page on the simple order-168 counterexample).
Kantor discusses Mitchell's work in some detail, but does not mention the
simple order-168 group explicitly.

CHAPTER V

THE KALEIDOSCOPE

"This is an account of the discrete groups generated by reflections…."

— Regular Polytopes , by H.S.M. Coxeter (unabridged and corrected 1973 Dover reprint of the 1963 Macmillan second edition)

"In this article, we begin a theory linking hyperplane arrangements and invariant forms for reflection groups over arbitrary fields…. Let V  be an n-dimensional vector space over a field F, and let G ≤ Gl_n (F) be a finite group…. An element of finite order in Gl(V ) is a reflection if its fixed point space in V  is a hyperplane, called the reflecting hyperplane. There are two types of reflections: the diagonalizable reflections in Gl(V ) have a single nonidentity eigenvalue which is a root of unity; the nondiagonalizable reflections in Gl(V ) are called transvections and have determinant 1 (note that they can only occur if the characteristic of F is positive)…. A reflection group is a finite group G  generated by reflections."

— Julia Hartmann and Anne V. Shepler, "Reflection Groups and Differential Forms," Mathematical Research Letters , Vol. 14, No. 6 (Nov. 2007), pp. 955-971

"… the class of reflections is larger in some sense over an arbitrary field than over a characteristic zero field. The reflections in Gl(V ) not only include diagonalizable reflections (with a single nonidentity eigenvalue), but also transvections, reflections with determinant 1 which can not be diagonalized. The transvections in Gl(V ) prevent one from developing a theory of reflection groups mirroring that for Coxeter groups or complex reflection groups."

— Julia Hartmann and Anne V. Shepler, "Jacobians of Reflection Groups," Transactions of the American Mathematical Society , Vol. 360, No. 1 (2008), pp. 123-133 (Pdf available at CiteSeer.)

See also A Simple Reflection Group of Order 168 and this morning's Savage Logic.