Volume 9, Issue 3

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The Modular Group

# Conjugate Subgroups

The four examples of cyclic subgroups outlined in the previous subsection give a complete description of the four types of subgroups possible in the modular group: any subgroup of the modular group is conjugate to a subgroup generated by , , an iterate of , or an iterate of an element of the same type as .

Recall that in a group G, a subgroup H is conjugate to a subgroup K if there exists an element such that the entire subgroup H can be generated by computing for every . More compactly, we write .

Consider a function in the modular group that generates an order-three subgroup. A segment of the unit circle is included to help orient our view in the following animation.

Figure 10. The action of order-three f on a selection of fundamental regions.

The subgroup generator is conjugate equivalent to , that is, there is an element such that . The function will map the fixed point of onto , the fixed point of . Moreover, will map the cluster of fundamental regions attached to the fixed point of onto the cluster about .

Figure 11. The action of conjugator H.

Here is a side-by-side view of the actions of on the right and on the left.

Figure 12. The action of and f.

We mentioned earlier that a counterclockwise winding about the origin was associated with the left-to-right shifting of . The function that carries out the winding action is . The following animation shows how this composition of and produces the winding action. A skeleton view of the fundamental region cluster connected at the origin is shown in each frame to provide a better orientation. Once again, intermediate frames have been added to give the impression of continuous motion between the actual fundamental regions of the modular group.