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Symmetries of Columns
We define a column by a realvalued 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
20010129
