Chapter 14.8, Problem 6E

To determine

To state whether the group in exercises 3 and 4 are commutative or not.

Answer to Problem 6E

Yes, the group in exercises 3 and 4 are commutative.

Explanation of Solution

Given information: A symmetry group is called commutative if $A\circ B=B\circ A$ for every pair of symmetry A and B in the group.

The symmetry group of the table as shown 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	$H_0$	$R_k$
$R_k$	$R_k$	$H_0$	I	${\text{R}}_j$
$H_0$	$H_0$	$R_k$	${\text{R}}_j$	I

The table of exercise 3 are as follow,

$°$	I $R_{0,}$ ${}_{120}$ $R_{0,}$ ${}_{240}$
I $R_0$ ${}_{120}$ $R_0$ ${}_{240}$	I $R_0$ ${}_{120}$ $R_0$ ${}_{240}$ $R_0$ ${}_{120}$ $R_0$ ${}_{240}$ I $R_0$ ${}_{240}$ I $R_0$ ${}_{120}$

The table of exercise 4 are as follow,

$°$	I $R_{0,}$ ${}_{90}$ $R_{0,}$ ${}_{180}$ $R_{0,}$ ${}_{270}$
I $R_0$ ${}_{90}$ $R_0$ ${}_{180}$ $R_0$ ${}_{270}$	I $R_0$ ${}_{90}$ $R_0$ ${}_{180}$ $R_0$ ${}_{270}$ $R_0$ ${}_{90}$ $R_0$ ${}_{180}$ $R_0$ ${}_{270}$ I $R_0$ ${}_{180}$ $R_0$ ${}_{270}$ I $R_0$ ${}_{120}$ $R_0$ ${}_{270}$ I $R_0$ ${}_{90}$ $R_{0,}$ ${}_{270}$

From the above table the commutative rule is applicable,

Therefore,

Thegroup in exercises 3 and 4 are commutative.