4.4.2 The Group of Meromorphic Automorphisms on
It is generally common to consider a meromorphic function as a function in the form of . Let denote the group of meromorphic automorphisms on .
To make more profound conclusions on the structure of , we will introduce certain concepts from group theory.
Definition 4.4.1 (Coset).
Let be a group, and let be a subgroup (operation denoted by juxtaposition). Then the left coset of in with respect to is defined as
The right coset is defined as
The subgroup is normal iff the left and right cosets are equal. The notation is used to represent a normal subgroup. Cosets, like groups and sets, are unordered.
Theorem 4.4.4.
Let be a group and a subgroup. The set of left cosets
admits a group structure with operation
if and only if is a normal subgroup of (i.e. ).
Proof.
We prove the two implications separately.
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If then is a group.
Assume is normal, , so for every (equivalently ). Define a product on by
We now verify that this product is well-defined: if and then we need . Since , there exists with , and since there exists with . Then
Because is normal we have , and , so . Hence , meaning that . Thus the product is well-defined.
Associativity follows from associativity in :
The identity is , since and similarly on the other side. The inverse of is , because . Thus is a group.
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If can be given a group structure via the coset multiplication, then .
Fix and arbitrarily. By assumption, we have
Because , we also have
implying that , and hence for any , . Hence, . Now replacing with and rearranging yields , or that . Therefore, is normal.
Under the normality of , the group is known as the quotient group of by .
Remark.
Every subgroup of an abelian group is normal.
Definition 4.4.2 (Group Homomorphism).
Let and be groups. A function is said to be a group homomorphism if
Definition 4.4.3 (Group Isomorphism).
A group homomorphism is called an isomorphism if it is bijective.
If there exists an isomorphism between two groups and , they are said to be isomorphic, denoted by . The utility of groups allows us to classify them according to their structure: if two groups are isomorphic, they are essentially the same from a group-theoretic perspective. This viewpoint lets us replace complicated groups with simpler, isomorphic ones, and study their properties without loss of generality.
Let us now examine . Let such that . It follows that maps to bijectively and . Therefore, has the form , where and are constants.
Let such that . Then,
is in and . By the property above, for some complex and nonzero . Hence,
Let and . Then
In this specific construction, . Let the matrix correspond to this transformation, where for any nonzero scalar , corresponds to . Therefore, we can arbitrarily pick to be .
Therefore, there exists a one-to-one correspondence between and the group under matrix multiplication of
The quotient group is taken because the matrix corresponds to the same transformation as . This group, denoted by
is known as the projective special linear group of order , and is isomorphic to the group of Möbius transformations, consisting of all complex linear fractional transformations.
Therefore, any meromorphic automorphism on is a composition of rotations, dilations, translations, and inversions. We will now state this formally:
Theorem 4.4.5 (The Meromorphic Automorphism Group on ).
, is a Möbius transformation. In other words, satisfying such that
Moreover, every such Möbius transformation is in .
The group of holomorphic automorphisms on , or , is also a subgroup of .
Proposition 4.4.1.
Suppose we have two Möbius transformations represented by the matrices and . Then their composition is a Möbius transformation represented by .
Proof.
From simple substitution, we have
which corresponds to the product .
We have now introduced three of the most important regions in complex analysis: , , and . Their importance will be later explained by the Uniformization Theorem (@ thm:uniformization).