Groups: Properties of groups

Definition and Basic Concepts of a Group

A group is a mathematical structure that combines an operation with a set of elements, subject to certain conditions.
It contains a set of elements and an operation that combines any two of the elements to form a third element within the set.
A group is well-defined, meaning that performing the operation on any two elements in the set will always produce a result that is itself an element in the set.
The operation must be associative, that is, for any elements a, b, and c in the group, the operation satisfies (a • b) • c = a • (b • c).

Identity Element: There is an element, commonly denoted by ‘e’ called the identity element, that when combined with any other element using the group operation, leaves the other element unchanged. Mathematically, for any element ‘a’ in the set, we can write: e • a = a • e = a.
Inverse: Each element in the group has an inverse. This is another element which, when combined with the original using the group operation, results in the identity element. If ‘a’ is an element in the group and ‘a^(-1)’ is its inverse, we can express this as: a • a^(-1) = a^(-1) • a = e.

The integers, with the operation being addition, form a group. 0 is the identity element, and the inverse of any integer n is the integer -n.
The non-zero real numbers, with the operation being multiplication, form a group. The number 1 is the identity element, and the inverse of any non-zero real number r is the reciprocal of number 1/r.

A subgroup is a set of elements taken from a larger group that also obey the group properties.
To verify whether a subset H of a group G is a subgroup, it is generally enough to confirm that the identity is in H, the product of any two elements from H is still in H, and the inverse of any element in H is still in H.

Two groups are isomorphic if there is a way to pair up the elements of the two groups such that the group operations are preserved.
This is defined more formally by a bijective function f from group G to group H such that for all a and b in G, f(a •_G b) = f(a) •_H f(b), where •_G and •_H denote the group operations in G and H respectively. The function f in this case is called an isomorphism.
If two groups are isomorphic, they are essentially the ‘same’ group, dressed up in different clothes.+