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In this section, we classify the closed subgroups of up to scaling and conjugacy in . Also, we compute the normalizers of these subgroups in .
Lemma 2.1 Suppose that C is a compact subgroup of . Then .
Proof: If and , then generates a noncompact subgroup of (isomorphic to ). It follows that .
Proposition 2.2 Suppose that G is a closed connected subgroup of . Then, up to conjugacy and scaling, G is one of the subgroups
Proof: If , then connectivity implies that . If , then connectivity implies that G is group isomorphic to either or . In the first case, it follows from Lemma 2.1 that . In the second case, there is a smooth isomorphism . This isomorphism is given by for some (defined as h(1)). By assumption . If , then . If , then by axial scaling we can arrange that and .
>From now on, we use the abbreviations
The proper closed subgroups of
are given by ,
the subgroup of rotations of the
cylinder through angles which are multiples of .
to be the subgroup of unit axial translations of the
cylinder generated by the element
Finally, for any
Of course, .
Proof: Since is abelian, we can write where C is compact and . Clearly, . By Lemma 2.1, or .
Assume that . Since is connected, the only subgroup satisfying is . Suppose next that . We claim that or . Choose the smallest positive such that there is with . Since , it follows that , where is the subgroup of generated by (0,t). By making an axial scaling, we can set t=1 so that .
Now assume that
then it follows from
Proposition 2.2 that
In the latter
case, we can choose a generator
with smallest b>0.
Making an axial scaling, we can suppose that the generator is of the form
In other words,
so we can suppose that
Using formula (2.1) we compute that
where is an abbreviation for . Hence up to conjugacy, we may suppose that . The case is the distinguished case .
Proposition 2.4 The normalizers of the subgroups have the form
where the subgroup is as given in Table 1.
Proof: Since is abelian, it is clear that . Hence for some subgroup . We compute that is the element . Hence, H consists of those elements that preserve . The element acts as -I on and so is always contained in H. It follows that or . It now suffices to determine whether or not preserves , that is, whether or not is preserved by the transformation .
Next: Untwisted Symmetry Groups Up: Symmetries of Columns Previous: Symmetries of Columns Marty Golubitsky