# Groups: Abstract groups

## 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, (a*b)*c equals a*(b*c).**Identity**: There exists an element e in G such that for all elements a in G, e*a = a*e = a.**Invertibility**: For every element a in G, there exists an inverse element a’ in G such that a*a’ = 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.