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# Symmetries of Columns

We define a column by a real-valued function f on the cylinder . Let . The function measures the height of the column in the direction normal to the cylinder at the point .

The group of symmetries of the cylinder is

where acts on by

Multiplication in follows from the definition of the action. Suppose that is in for j=1,2, where , and . Then multiplication is given by

 (1)

We wish to classify columns by their symmetries. A symmetry of the column is such that

The symmetry group is the collection of all symmetries of f. We classify all subgroups which are symmetry subgroups for some column f.

Our classification proceeds as follows. To each subgroup , we can associate the normal subgroup

 (2)

(So consists of the pure `translations' in .) Thus it suffices to
(i)
classify the closed subgroups of ,
(ii)
for each subgroup in (i), compute the subgroups that satisfy (2.2).
The calculation in (ii) is simplified by observing that is contained in the normalizer of .

As usual, we identify conjugate subgroups of . In addition, we identify subgroups that are related by axial scalings. More precisely, we define the scaling transformation by

Provided , this is an isomorphism. We say that two subgroups , are related by a scaling if for some nonzero .

Next: Classification of Subgroups of Up: A Symmetry Classification of Previous: Introduction
Marty Golubitsky
2001-01-29