Chapter 14.8, Problem 9E

a.

To determine

To findthe four lines symmetries of the square form a subgroup.

Answer to Problem 9E

The four lines symmetries of the square do not form a subgroup, as there is no identity.

Explanation of Solution

Given information:

The figure of a square.

The figure of square has the four rotational symmetry. Each symmetry has center O and rotates the figure onto itself. Note that $90°$ rotational symmetry is another name for point symmetry.

  $\begin{array}{l}\left(1\right)\text{ 9}0°{\text{ rotational symmetry: R}}_{0,90}\\ \left(1\right)180°{\text{ rotational symmetry: R}}_{0,180}\\ \left(2\right)270°{\text{ rotational symmetry: R}}_{0,270}\\ \left(3\right)360°\text{ rotational symmetry: I is the identity}\text{.}\end{array}$

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

The group table is given below,

$°$	I	${\text{R}}_j$	$R_k$	$H_0$
I	I	${\text{R}}_j$	$R_k$	$H_0$
${\text{R}}_j$	${\text{R}}_j$	I	$H_0$	I
$R_k$	$R_k$	$H_0$	I	${\text{R}}_j$
$H_0$	$H_0$	$R_k$	${\text{R}}_j$	I

Therefore, the four lines symmetries of the square do not form a subgroup, as there is no identity.

b.

To determine

The find that the symmetry group of the equilateral triangle have a subgroup.

Answer to Problem 9E

The group is commutative as it is an abelian group and symmetry group of the equilateral triangle can form a subgroup.

Explanation of Solution

Given:

The figure of equilateral triangle.

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 commutative as it is an abelian group and symmetry group of the equilateral triangle can form a subgroup.

c.

To determine

The find that the two symmetries of the figure form a subgroup.

Answer to Problem 9E

The two symmetries of the figure form a subgroup that are I and $H_0$ .

Explanation of Solution

Given:

The figure has 4 rotational symmetry.

Concept Used:

The group is commutative if it has four property,

The product of two symmetry is another symmetry.

The set of symmetries contains the identity.

Each symmetry has an inverse that is also a symmetry.

Forming of product is an associative group,

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

Therefore, the two symmetries of the figure form a subgroup that are I and $H_0$ .