# Chemistry - How are there two C3 rotation axes in ammonia?

## Solution 1:

In a character table for $C_{3v}$, $ 2C_3$, means that the element is actually 2 $C_3$ elements in the same **conjugacy class**. That means the two $C_3$ operations are related in some way.

**In fact, it means they share the same single axis.** One $C_3$ operation is a $120^\circ$ turn around the axis, and the other $C_3$ operation is a $240^\circ$ turn around the same axis in the same direction - or equivalently a $120^\circ$ turn around the same axis in the opposite direction.

In the image below the two elements $C_3$ and $C_3^{-1}$ are the two symmetry elements of the $2C_3$ class, which are rotations about the same axis out of the plain of the image.

Symmetry elements of the $C_{3v}$ point group Image taken from Fig S5 in supplementary information of T. Karin, et al, Phys. Rev. B 94, 041201

## Solution 2:

The top row of the character table does not refer to *symmetry elements*, but rather *symmetry operations*. (For the difference between the two, see Wikipedia:Molecular symmetry.)

In this particular case, the difference is that there is only one $C_3$ symmetry element in ammonia (i.e. the rotation axis), but there are two operations: one corresponds to the clockwise rotation, and the other to the anticlockwise rotation (which is equivalent to two clockwise rotations, should you prefer that).

It is the *symmetry operations*, not the *elements*, which form the so-called "group" in group theory (see Wikipedia:Group (mathematics)).

In mathematics, a group is a set equipped with

a binary operationthat combines any two elements to form a third element in such a way that three conditions called group axioms are satisfied, namelyassociativity, identity and invertibility.

(Emphasis mine.) The "binary operation" for *symmetry operations* is to simply perform one operation after another, which is somewhat akin to composition. Note that the word "element" in the quoted paragraph refers to an abstract member of the group: it is mildly unfortunate that the elements are *symmetry operations*, not symmetry elements.