# Regular Triangle Symmetry Group Exploration

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**Objective:**
Understanding the finite symmetry groups.

## The square

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

<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> |

## The equilateral triangle

Analyze the symmetry group *D3* of the equilateral triangle:

- How many elements are in this group?
- 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>?
- 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>?
- How do rotations behave?
- Can you spot
*C3*as a subgroup of*D3*? What is it? - Find all subgroups.
- Write out a multiplication table for
*D3*.

**Handin:**
A sheet with answers to all questions.