Chapter 14.8, Problem 7E

a.

To determine

To describe all the rotational symmetries of the figure equilateral triangle.

Answer to Problem 7E

The group table for six symmetriesare given below,

$°$	I	${\text{R}}_{0,120}$	${\text{R}}_{0,240}$
I	${\text{R}}_{0,120}$	${\text{R}}_{0,120}$	${\text{R}}_{0,240}$
${\text{R}}_{0,240}$	${\text{R}}_{0,240}$	I	I

Explanation of Solution

Given information:

The figure of an equilateral triangle.

The figure of equilateral triangle has the three rotational symmetry. Each symmetry has center O and rotates the figure onto itself. Note that $120°$ rotational symmetry is another name for point symmetry.

  $\begin{array}{l}\left(1\right)120°{\text{ rotational symmetry: R}}_{0,120}\\ \left(2\right)240°{\text{ rotational symmetry: R}}_{0,240}\\ \left(3\right)360°\text{ rotational symmetry: I is the identity}\text{.}\end{array}$

The identity mapping always maps a figure onto itself, we usually include the identity when listing the symmetries of a figure.

The group table for six symmetries are given below,

$°$	I	${\text{R}}_{0,120}$	${\text{R}}_{0,240}$
I	${\text{R}}_{0,120}$	${\text{R}}_{0,120}$	${\text{R}}_{0,240}$
${\text{R}}_{0,240}$	${\text{R}}_{0,240}$	I	I

b.

To determine

The find that the group is not commutative.

Answer to Problem 7E

The group is not commutative as it is not a non-abelian group.

Explanation of Solution

Given:

The figure equilateral triangle has 3 rotational symmetry.

Concept Used:

The group is commutative if it has four property,

1. The product of two symmetry is another symmetry.
2. The set of symmetries contains the identity.
3. Each symmetry has an inverse that is also a symmetry.
4. Forming of product is an associative group,

  $A°\left(B°C\right)=\left(A°B\right)°C$ for any three group.

Therefore, the group is not commutative as it is not abelian group.