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 (*) :
An Abelian group is a group that is follows the axioms 1 – 4 with the addition of one property:
For the remainder of this post, we will explore these axioms and look at some examples
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.
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.
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
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
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 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