Special linear group:SL(2,Z)
This article is about a particular group, i.e., a group unique upto isomorphism. View specific information (such as linear representation theory, subgroup structure) about this group
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The group is defined as the group, under matrix multiplication, of matrices over , the ring of integers, having determinant .
In other words, it is the group with underlying set:
The group also has the following equivalent descriptions:
- The inner automorphism group of braid group:B3, i.e., the quotient of by its center.
- The amalgamated free product of cyclic group:Z4 and cyclic group:Z6 over amalgamated subgroup cyclic group:Z2 (living as Z2 in Z4 and Z2 in Z6 respectively).
Thinking of as a group of matrices, we see that it is an example of an arithmetic group.
|order||infinite (countable)|| The group is infinite because, for instance, it contains all matrices of the form for . |
As a set, the group is contained in the set of all matrices over . This can be identified with , which is countable since is countable. Thus, is also countable.
|exponent||infinite||The group contains the element , which has infinite order.|