# Regular Triangle Symmetry Group Exploration

Objective: Understanding the finite symmetry groups.

## The square

Complete the multiplication table for D4, the symmetry group of the square.

[itex]E[/itex] (identity) [itex]R[/itex] (rotation 90) [itex]R^2[/itex] (rotation 180) [itex]R^3[/itex] (rotation 270) [itex]M1[/itex] (reflection) [itex]M2[/itex] (reflection) [itex]M3[/itex] (reflection) [itex]M4[/itex] (reflection)
[itex]E[/itex]
[itex]R[/itex]
[itex]R^2[/itex]
[itex]R^3[/itex]
[itex]M1[/itex]
[itex]M2[/itex]
[itex]M3[/itex]
[itex]M4[/itex]

## The equilateral triangle

Analyze the symmetry group D3 of the equilateral triangle:

1. How many elements are in this group?
2. What is [itex]M1[/itex] x [itex]M1[/itex] = [itex]M1^2[/itex]? , [itex]M2[/itex] x [itex]M2[/itex] = [itex]M2^2[/itex]? , [itex]M3[/itex] x [itex]M3[/itex] = [itex]M3^2[/itex]?
3. What is [itex]M1[/itex] x [itex]M2[/itex]? , [itex]M2[/itex] x [itex]M1[/itex]? , [itex]M3[/itex] x [itex]M1[/itex]? , [itex]M1[/itex] x [itex]M3[/itex]? , [itex]M3[/itex] x [itex]M2[/itex]? , [itex]M2[/itex] x [itex]M3[/itex]?
4. How do rotations behave?
5. Can you spot C3 as a subgroup of D3? What is it?
6. Find all subgroups.
7. Write out a multiplication table for D3.

Handin: A sheet with answers to all questions.