Properties of groups

Definition and Basic Properties of Groups

A group is a mathematical structure comprised of a set of elements along with an operation that combines a pair of elements to produce a third.
Within the group, this operation must be associative. In other words, for any three elements a, b, and c in the group, (a * b) * c = a * (b * c).
There must be an identity element denoted as ‘e’ in the group. Any element ‘a’ combined with ‘e’ via the group operation results in ‘a’. Formally, for any element ‘a’ in the group, a * e = e * a = a.
Every element ‘a’ in the group must have an inverse element ‘a^-1’ such that when ‘a’ and ‘a^-1’ are combined with the group operation, the result is the identity element. Formally, for every element ‘a’ in the group, there exists ‘a^-1’ in the group such that a * a^-1 = a^-1 * a = e.

A group is called abelian or commutative if the order of operation does not matter. For any elements ‘a’ and ‘b’ in the group, a * b = b * a.
If the operation order matters and a * b ≠ b * a for at least one pair of elements ‘a’ and ‘b’ in the group, the group is classified as non-abelian.

A subgroup of a group is a subset of the group that is also a group with respect to the operation of the larger group.
For a subset ‘H’ of a group ‘G’ to be a subgroup, it must be non-empty, it must be closed under the group operation, and every element of ‘H’ must have its inverse in ‘H’.
The order of a subgroup is equal to the number of its elements.

Every element ‘a’ in a group ‘G’ generates a cyclic subgroup of ‘G’. This subgroup consists of the identity, the element ‘a’, its powers, and its inverse.
A group is cyclic if it can be generated by a single element, meaning all its elements are powers of one element.

Symmetry groups are groups consisting of symmetry transformations.
The order of a symmetry group is the number of symmetries it has.
Each group element corresponds to a particular symmetry of the object. The group operation corresponds to performing first one symmetry operation and then another.
An example of a symmetry group is the rotation group of a square, which consists of rotations of multiples of 90° only. Another example is the flip group of a square, consisting of a flip about the vertical axis, a flip about the horizontal axis, and the absence of a flip.