Groups: Abstract groups

Understanding Abstract Groups

An abstract group is a set, along with an operation, that satisfies four properties: closure, associativity, identity and invertibility.
A set G, combined with an operation *, forms a group if the following hold:
- Closure: For all elements a, b in G, a*b also belongs to G.
- Associativity: For all elements a, b, c in G, (ab)c equals a(bc).
- Identity: There exists an element e in G such that for all elements a in G, ea = ae = a.
- Invertibility: For every element a in G, there exists an inverse element a’ in G such that aa’ = a’a = e, where e is the identity element.

An example of an abstract group is (Z, +), the set of integers under the operation of addition.
Another example is the set of non-zero real numbers (R\{0}, *) under the operation of multiplication.

If there is an element a in a group G, such that every other element in the group can be written as a power of a, then the group is known as a cyclic group and a is called a generator of the group.
For example, the group (Z, +) is cyclic with generator 1 or -1. Any integer can be written as a sum of 1’s or -1’s.

A subset H of a group G is called a subgroup of G if H itself is a group under the operation of G.
The trivial subgroup of any group G is the subgroup that consists only of the identity element of G.

The order of a group is the number of elements in the group.
The order of an element in a group is the smallest positive integer n such that a^n = e, where a is the element, n is an integer, and e is the identity element of the group.

Abstract Groups are the fundamental objects studied in the field of abstract algebra.
They have significant applications in fields such as physics, cryptography, and theoretical computer science.