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A Simple Reflection Group of Order 168

by Steven H. Cullinane,
Dec. 7, 2007-Nov. 27, 2012

"Let G be a finite, primitive subgroup of GL(V) = GL(n,D), where V is an n-dimensional vector space over the division ring D. Assume that G is generated by 'nice' transformations. The problem is then to try to determine (up to GL(V)-conjugacy) all possibilities for G. Of course, this problem is very vague. But it is a classical one, going back 150 years, and yet very much alive today."

-- William M. Kantor, "Generation of Linear Groups," pp. 497-509 in The Geometric Vein: The Coxeter Festschrift, published by Springer, 1981

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

More precisely (update of 19 Oct. 2010):

"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 ≤ Gln (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.)

A definition from 1955:

"An invertible linear transformation of a finite dimensional vector space V over a field K will be called a reflection if it is of order two and leaves a hyperplane pointwise fixed. A group G of linear transformations of V is a finite reflection group if it is a finite group generated by reflections." --Claude Chevalley, "Invariants of Finite Groups Generated by Reflections," American Journal of Mathematics, Vol. 77, No. 4 (Oct., 1955), pp. 778-782

This note points out that the above definitions include 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. 

The Eightfold Cube

The eightfold cube
Set of three reflections generating the simple group of order 168
Generating reflections
For details, see The Fano Plane Revisualized and Binary Coordinate Systems. For a check of the proof in the latter, see a MAGMA calculation.

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.

Update of Feb. 15, 2010--

Background: Coxeter Groups

Theorems 15.1 and 15.2 of Alexandre Borovik's book (1st ed. Nov. 10, 2009)
Mirrors and Reflections: The Geometry of Finite Reflection Groups

15.1 (p. 114): Every finite reflection group is a Coxeter group.

15.2 (p. 114): Every finite Coxeter group is isomorphic to a finite reflection group.

Consider in this context the above simple reflection group of order 168.

(Recall that "…there is only one simple Coxeter group (up to isomorphism); it has order 2…" –A.M. Cohen.)

Update of Feb. 18, 2010--

Truth, Geometry, Algebra

1. According to H.S.M. Coxeter and Richard J. Trudeau

"There is a pleasantly discursive treatment of Pontius Pilate's unanswered question 'What is truth?'."

– Coxeter, 1987, introduction to Trudeau's The Non-Euclidean Revolution

1.1 Trudeau's Diamond Theory of Truth
1.2 Trudeau's Story Theory of Truth

2. According to Alexandre Borovik and Steven H. Cullinane

2.1 Coxeter Theory according to Borovik

2.1.1 The Geometry–
          Mirror Systems in Coxeter Theory
2.1.2 The Algebra–
          Coxeter Languages in Coxeter Theory

2.2 Diamond Theory according to Cullinane

2.2.1 The Geometry–
           Examples: Eightfold Cube and Solomon's Cube
2.2.2 The Algebra–
           Examples: Cullinane and (rather indirectly related) Gerhard Grams

Summary of the story thus far:

Diamond theory and Coxeter theory are to some extent analogous-- both deal with reflection groups and both have a visual (i.e., geometric) side and a verbal (i.e., algebraic) side.  Coxeter theory is of course highly developed on both sides. Diamond theory is, on the geometric side, currently restricted to examples in at most three Euclidean (and six binary) dimensions. On the algebraic side, it is woefully underdeveloped. For material related to the algebraic side, search the Web for generators+relations+"characteristic two" (or "2") and for generators+relations+"GF(2)". (This last search is the source of the Grams reference in 2.2.2 above.)

Update of Feb. 21, 2010--


From the Wikipedia article "Reflection Group" that I created on Aug. 10, 2005as revised on Nov. 25, 2009

Historically, (Coxeter 1934) proved that every reflection group [Euclidean, by the current Wikipedia definition] is a Coxeter group (i.e., has a presentation where all relations are of the form ri2 or (rirj)k), and indeed this paper introduced the notion of a Coxeter group, while (Coxeter 1935) proved that every finite Coxeter group had a representation as a reflection group [again, Euclidean], and classified finite Coxeter groups.

Finite fields

This section requires expansion.

When working over finite fields, one defines a "reflection" as a map that fixes a hyperplane (otherwise for example there would be no reflections in characteristic 2, as −1=1 so reflections are the identity). Geometrically, this amounts to including shears in a hyperplane. Reflection groups over finite fields of characteristic not 2 were classified in (Zalesskiĭ & Serežkin 1981).

Related material:

This note, "A Simple Reflection Group of Order 168," and

"Determination of the Finite Primitive Reflection Groups over an Arbitrary Field of Characteristic Not 2,"

by Ascher Wagner, U. of Birmingham, received 27 July 1977

Journal   Geometriae Dedicata
Publisher   Springer Netherlands
Issue   Volume 9, Number 2 / June, 1980

Ascher Wagner's 1977 dismissal of reflection groups over fields of characteristic 2

[A primitive permuation group preserves
no nontrivial partition of the set it acts upon.]

Clearly the eightfold cube is a counterexample.

Update of Feb. 24, 2010--


A topic related to the above material on reflection groups over GF(2)--

Transvection groups over GF(2). See, for instance...
  1. Binary Coordinate Systems, by Steven H. Cullinane, 1984

  2. Classification of the Finite N-Generator Transvection Groups Over Z2, by Jizhu Nan and Jing Zhao, 2009, Advances in Applied Mathematics Vol. 44 Issue 3 (March 2010), 185–202

  3. Anne Shepler, video of a talk on Nov. 4, 2004, "Reflection Groups and Modular Invariant Theory"
Update of July 8, 2010--

Coxeter vs. Fano

Excerpts from Coxeter's Projective Geometry sketch his attitude toward geometry in characteristic two.

Update of November 27, 2012—

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 (Nov. 27, 2012) 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 .

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 above.  Kantor discusses Mitchell's work in some detail, but does not mention the simple order-168 group explicitly.

For reflection groups over other finite fields, see the following:

1. Linear Groups Generated by Reflection Tori (pdf), by Arjeh M. Cohen, Hans Cuypers, and Hans Sterk (Canad. J. Math. Vol. 51 (6), 1999, pp. 1149-1174).

2. From a page for Mathematics of the USSR-- Izvestiya:

by A. E. Zalesskii and V. N. Serezhkin

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.
Classification  AMS MSC: Primary: 20H15, 51F15
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