To be used in future
 Chapter 3: Symmetries Up Chapter 3: Symmetries Section 3.2: Examples of Symmetries 

3.1 Why symmetries?

Symmetries play an important role in classifying and identifying structures we find in the real world, such as many patterns we find in nature or in things we create. These are often examples of discrete symmetries↓ (say reflections and fixed angle rotations). These combine to form the “space groups↓”, which you may have been introduced to in condensed matter physics.
On the other hand, many of the microscopic laws of physics have ↓ [A]  [A] i.e., symmetry operation depending on one or more continuous variables, such as translations, see below., which in this case can usually be linked to the freedom to choose and change reference frame in space and/or time. (Remember your relativity?) The symmetry then states that the physics is independent of the choice. We shall try to link such independence into an invariance under the transformations associated with a symmetry, which will prove quite useful in quantum problems.
The typical mathematical structure of the set of all symmetry transformations is a “group”↓—usually an even more specialised structure called a Lie group↓; I shall not discuss these concepts here, but note that we can deduce quite a bit of the physical properties of a system from the mathematical theory of the underlying symmetry group.

3.1.1 Linearity↓

We shall only look at linear symmetry transformations. There is no a priori reason to exclude non-linear ones, but since most normal space-time symmetries are linear, and since most of the basic laws of physics are linear, as we shall see below, there is a natural bias towards this class of transformations. Of course quantum mechanics, where linear operators act in a linear vector space (the Hilbert space) is not easy to combine with non-linear transformations.
 Chapter 3: Symmetries Up Chapter 3: Symmetries Section 3.2: Examples of Symmetries