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Symmetry, Automorphisms, and
Visual Group Theory

Steven H. Cullinane, March 2, 2010

From the conclusion
of Weyl's Symmetry

One example of Weyl's "structure-endowed entity" is a partition of a six-element set into three disjoint two-element sets– for instance, the partition of the six faces of a cube into three pairs of opposite faces.

The automorphism group of this faces-partition contains an order-8 subgroup that is isomorphic to the abstract group C2×C2×C2 of order eight--

Order-8 group generated by reflections in three midplanes of cube

The action of Klein's simple group of order 168 on the Cayley diagram of C2×C2×C2 in the following note on Carter's 2009 book Visual Group Theory furnishes an example of Weyl's statement that

"… one may ask with respect to a given abstract group: What is the group of its automorphisms…?"

Visual Group Theory
Note of March 2, 2010

The current article on group theory at Wikipedia has a Rubik's Cube as its logo– 

Wikipedia article 'Group theory' with Rubik Cube and quote from Nathan Carter-- 'What is symmetry?'


The article quotes Nathan C. Carter on the question "What is symmetry?"

This naturally suggests the question "Who is Nathan C. Carter?"

A search for the answer yields the following set of images…

Labelings of the eightfold cube

Click image for some historical background.

Carter turns out to be a mathematics professor at Bentley University.  His logo– an eightfold-cube labeling (in the guise of a Cayley graph)– is in much better taste than Wikipedia's.