# Sir i need help for understanding symmetry elements which is the basis of group theory.i have read many books containing this subject , however i can't imagine these elements.please help me.thanking you?

##### 1 Answer

Many good group theory texts *should* have images... but **this website** is great for additional visualization practice. Bookmark it! You can perform the operations by clicking the button next to the symmetry element.

**Also, you will find this website useful later**; it'll help you check your reduced reducible representations, so keep this website in mind as well. For instance,

**Symmetry operations** can be categorized in general as:

- Identity,
#hatE# , symmetry element =#E# (nothing) - Rotation,
#hatC_n# , symmetry element =#C_n# (an axis) - Reflection,
#hatsigma# , symmetry element =#sigma# (a plane) - Inversion,
#hati# , symmetry element =#i# (a dot)

We can use

*Note:* *is the symmetry operation, and* *is its symmetry element.*

**IDENTITY**

The **identity operation**

There really is no point in identifying what the **symmetry element**

**ROTATION**

The **rotation operation**, **symmetry element** is the

For example,

When you rotate

From this angle, it is more noticeable that you can rotate **three-fold rotation axis** *operation*.

**REFLECTION**

The **reflection operation**, **symmetry element** is the plane itself.

#sigma_v# is*colinear*with the principal#C_n# axis (of the highest#n# ), and*lines up with*an outer atom.#sigma_h# is perpendicular to the principal#C_n# axis.#sigma_d# *bisects*two outer atoms, crossing through the center of the molecule, and is in between two#sigma_v# planes. It must not directly line up with an outer atom (otherwise it is#sigma_v# ).

Cyclobutane (

**ROTATION-REFLECTION (IMPROPER ROTATION)**

This is its own operation, **improper rotation** (rotation-reflection), but it really is just a combination of rotation and reflection in either order.

For example, *then* reflect through the horizontal plane (

You should convince yourself though that

**INVERSION**

The **inversion operation** **symmetry element**

The easiest way I can think of to describe it is that it takes the coordinates

Here's an example of inversion with a molecule that ** doesn't** have inversion symmetry, like

This is hard to visualize for molecules *with* inversion symmetry, because it looks like it does nothing. Practice with this one.