Working with Symmetry groups

Introduction to Symmetry Groups

A symmetry group is an algebraic structure that combines rules about symmetry and rotation.
Symmetry groups are essential tools for analysing the properties of geometrical objects, such as polygons or 3D polyhedra.
In mathematical terms, a symmetry group describes the symmetries that come from the structure of the object.
Examples of symmetry groups include reflectional symmetry, rotational symmetry and translational symmetry.

All symmetry groups have an identity, typically denoted as ‘e’. This represents a transformation that leaves the shape unchanged.
A transformation can be combined with another transformation by performing first one then the other. This operation is denoted by multiplication.
For instance, in a symmetry group, the set of transformations includes various rotations and reflection, depending on the structure of the object.
Every transformation has an inverse, which reverses the effect of the original transformation. Combining a transformation with its inverses results in the identity.

The order of a symmetry group is the number of distinct transformations it contains.
For instance, a square has an order of 8 because there are 4 rotations and 4 reflections that map the square onto itself.
The order of an element in a group is the smallest positive integer such that a certain operation on the element repeated ‘n’ times gives the identity.
For example, a 90° rotation of a square has order 4; after 4 such rotations, we arrive back at the original position.

The composition of transformations is an essential element of the study of symmetry groups. It involves performing two or more transformations consecutively.
There is a set order for composing transformations. If the transformations are represented as T and U, the expression TU means that first T is applied, followed by U.
However, the group operation for transformations is not commutative. In other words, TU may not be the same as UT. Such groups are known as non-Abelian groups.
To simplify calculations and to deduce the structure of a group, mathematical tables called Cayley tables are used.

A subgroup is a subset of a group that satisfies the group properties.
Subgroups retain the identity element of the main group and also include the inverses of all its members.
The order of a subgroup divides the order of the group.
Subgroups are useful for breaking down complex symmetry groups into simpler parts for analysis.