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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
follows from the definition of the action.
for j=1,2, where
Then multiplication is given by
We wish to classify columns by their symmetries. A symmetry of the
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
(So consists of the pure `translations' in .) Thus it suffices to
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
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