Group Multiplication Table

Let us consider the symmetry group of $NH_3$ molecule. These are:

• a 3-fold axis, associated with two symmetry operations: $C_3^+$ (+120$^o$ rotation) and $C_3^-$ (-120$^o$ rotation).
• 3 $\sigma_v$ vertical planes, $\sigma_v$, $\sigma_v'$, and $\sigma_v''$ associated with tree mirror reflections.

Now we add to these five symmetry operations the identity operator $E$ and show that all six symmetry operations joint a group. Particularly, it is easy to see that $C_3^+C_3^- = E$, where the identity operation $E$ can be considered as a "product" of the two rotation operators operations $C_3^+$ and $C_3^-$. It can also be seen that $\sigma_{va}C_3^+ = \sigma_{vb}$ and $C_3^+\sigma_{va} = \sigma_{vc}$. Following this procedure we can build the "multiplication table" presented below.

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Table 1: Multiplication Table for the $C_{3v}$ Group

	$E$	$C_{3}^+$	$C_3^-$	$\sigma_v$	$\sigma_v'$	$\sigma_v''$
$E$	$E$	$C_{3}^+$	$C_3^-$	$\sigma_v$	$\sigma_v'$	$\sigma_v''$
$C_3^+$	$C_3^+$	$C_3^-$	$E$	$\sigma_v'$	$\sigma_v''$	$\sigma_v$
$C_3^-$	$C_3^-$	$E$	$C_3^+$	$\sigma_v''$	$\sigma_v$	$\sigma_v'$
$\sigma_v$	$\sigma_v$	$\sigma_v''$	$\sigma_v'$	$E$	$C_3^-$	$C_3^+$
$\sigma_v'$	$\sigma_v'$	$\sigma_v$	$\sigma_v''$	$C_3^+$	$E$	$C_3^-$
$\sigma_v''$	$\sigma_v''$	$\sigma_v'$	$\sigma_v$	$C_3^-$	$C_3^+$	$E$

According to the Table 1, the "product" of each two symmetry transformations from six $E$, $C_{3}^+$, $C_3^-$, $\sigma_v$, $\sigma_v'$, and $\sigma_v''$ is equivalent to one of these transformations. It is clearly seen that the third and the fourth conditions of the group are also valid. Thus, these six operators build a group. This group is known as $C_{3v}$ group. The total number of operations in a group is called the group order. Therefore, the order of $C_{3v}$ is 6.

Let us consider the symmetry group of $H_2O$ molecule. The symmetry elements are:

• a 2-fold axis, associated with the symmetry operation: $C_2$ .
• two $\sigma_v$ vertical planes, $\sigma_v$, and $\sigma_v'$ associated with two mirror reflections.

Now we add to these four symmetry operations the identity operator $E$ and show that all six symmetry operations joint a group. The "multiplication table" presented below.

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Table 2: Multiplication Table for the $C_{2v}$ Group

	$E$	$C_2$	$\sigma_v$	$\sigma_v'$
$E$	$E$	$C_2$	$\sigma_v$	$\sigma_v'$
$C_2$	$C_2$	$E$	$\sigma_v'$	$\sigma_v$
$\sigma_v$	$\sigma_v$	$\sigma_v'$	$E$	$C_2$
$\sigma_v'$	$\sigma_v'$	$\sigma_v$	$C_2$	$E$

According to the Table 2, the "product" of each two symmetry transformations from six $E$, $C_2$, $\sigma_v$, and $\sigma_v'$ is equivalent to one of these transformations. It is clearly seen that the third and the fourth conditions of the group are also valid. Thus, these four operators build a group. This group is known as $C_{2v}$ group. The group order of $C_{2v}$ is 4.

Each point group is characterized by each own multiplication table.