**Binary operations** are operations such as addition, subtraction, multiplication, division, modulus etc. that are applied to two quantities.

example 1: is an example of an expression with addition as the binary operation

example 2: Let f and g be functions defined on sets A to B. Then the composition of the functions is a binary operation

We will use * to denote an arbitrary (general) binary operation.

A set G is “upgraded” into a **group** if it satisfied the following axioms under **one** binary operation (*) :

- Closure:
- Associativity:
- Identity:
- Inverse:

An **Abelian group** is a group that is follows the axioms 1 – 4 with the addition of one property:** **

- Commutativity:

For the remainder of this post, we will explore these axioms and look at some examples

Closure:

This means we can take any elements in the set G and perform the operation defined by * and the result will also be an element in the group.

Consider this example: If with regular addition, +, as the binary operation, then we want to know if we can take any two numbers in the set, add them, and get another number that is still in the set.

- (4 is in the set, hence condition is satisfied for this example)
- (2 is in the set, hence condition is satisfied for this example)

So far, it seems like G is closed under addition! If we can find at least one example of two numbers from G that, when added, result in a number that is not in the set, then we’ve shown that condition isn’t satisfied for all the elements of the set.

- which is not in the set. Hence, we can conclude that G is not closed under + since closure is not satisfied for ALL elements in the set!
- If we were trying to prove that G is a group under +, then we can already stop and conclude that it isn’t closed under the operation.

Associativity:

If a set G is associative, then we can take any three elements of a set G, perform the defined group operation, * , and the result will be the same irrespective of the order in which we applied the operation.

Consider the same example:

- Then So we can conclude that In other words, the placement of the brackets did not affect the result after the operation, +, was performed.
- Similarly,

If this holds for all elements in G, then we can conclude that G is associative!

- If the binary operation was division, for the same set G, then the placement of the brackets would be important. Let * denote division, then and . Hence the answer changed, so division is not associative for the set G

Identity:

i.e. there exists an element, called the identity element, such that for any element

To understand this, let’s look at some cases for different groups:

- under binary operation, +, has zero as the identity element. Since for any element in G, we can add zero and get that element:
- Set has 1 as the identity element since

Inverse:

i.e. here exists an element,

- Since has zero as the identity element. Then the inverse of
- Similarly, the inverse of and so on

Something worth noting for a group, under multiplication: the inverse law takes into account that 0 cannot have an inverse

So under some operation, an element and its inverse will give you the identity element!

- If we consider the set G under multiplication, then the inverse property is not satisfied since 0 has no inverse element. In other words, there is (where 1 is the identity element of G under multiplication).

We denote a set G that forms a group under a binary operation as

example 1: is a set G with addition as the binary operation

example 1: is a set G with multiplication as the binary operation

To show that a set is a group, we want to show that all 4 axioms are satisfied for the defined operation * simultaneously. If we can find at least one counter-example, then we can immediately conclude that a set is not a group.

In our case, G was not a group under addition since it wasn’t closed. All other properties were satisfied.

commutativity:

Commutativity tells us the order in which we apply the operation to our elements in some set doesn’t matter.

- For our example, under binary operation, +, we could add two elements, as and get the same result
- For instance, or . Note again, how 7 is not in the set, again reminding us that out set is not closed. In this case that our set was closed and commutative, G would be an Abelian group!

Now we’ll explore some sets that form groups and which don’t:

example 1: set of n x n matrices with real coefficients only form a group when every matrix is invertible. the identity matrix will be the identity element

example 2: the set of real numbers without zero forms a group under multiplication with identity 1. If zero is included in the set, then 0 does not have an inverse, hence inverse property isn’t satisfied and set under multiplication is not a group

example 3: the set of rational numbers, under addition with 0 as the identity forms a group

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