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## Classification of Subgroups of .

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

where

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 and . The proper closed subgroups of are given by , : the subgroup of rotations of the cylinder through angles which are multiples of . In addition, we set to be the subgroup of unit axial translations of the cylinder generated by the element . Finally, for any , we define

Of course, .

Theorem 2.3   Up to axial scaling and conjugacy, the closed subgroups are listed in Table 1.

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 . If , then it follows from Proposition 2.2 that or . If , then either or . 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 for some . In other words, . Note that , 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
2001-01-29