Next: Untwisted Symmetry Groups Up: Symmetries of Columns Previous: Symmetries of Columns 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 |