# 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.

## Examples of Abstract Groups

• 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.

## Cyclic Groups

• 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.

## Subgroups

• 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 and Elements

• 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.

## Importance of Abstract Groups

• 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.