Regular Triangle Symmetry Group Exploration

Objective: Understanding the finite symmetry groups.

1. Complete the multiplication table for D4

	<math>E</math> (identity)	<math>R</math> (rotation 90)	<math>R^2</math> (rotation 180)	<math>R^3</math> (rotation 270)	<math>M1</math> (reflection)	<math>M2</math> (reflection)	<math>M3</math> (reflection)	<math>M4</math> (reflection)
<math>E</math>
<math>R</math>
<math>R^2</math>
<math>R^3</math>
<math>M1</math>
<math>M2</math>
<math>M3</math>
<math>M4</math>

Analyze the symmetry group D3 of the equilateral triangle

2. How many elements are in this group?

3. What is <math>M1</math> x <math>M1</math> = <math>M1^2</math>? , <math>M2</math> x <math>M2</math> = <math>M2^2</math>? , <math>M3</math> x <math>M3</math> = <math>M3^2</math>?

4. What is <math>M1</math> x <math>M2</math>? , <math>M2</math> x <math>M1</math>? , <math>M3</math> x <math>M1</math>? , <math>M1</math> x <math>M3</math>? , <math>M3</math> x <math>M2</math>? , <math>M2</math> x <math>M3</math>?

5. How do rotations behave?

6. Can you spot C3 as a subgroup of D3? What is it?

7. Find all subgroups.

8. Write out a multiplication table for D3.

Handin: A sheet with answers to all questions.