Group

I was a math major in college. The first subject I learned in abstract algebra is Group. It was a very simple entity.

A Group is a tuple (𝐆, ⊕) of a set 𝐆 and a binary operation ⊕ on the set that satisfy the following properties:

• Associativity: (𝑎⊕𝑏)⊕𝑐 = 𝑎⊕(𝑏⊕𝑐) for all 𝑎, 𝑏, c ∈ 𝐆
• Identity element: There is an element 𝑒 in 𝐆 such that 𝑎⊕𝑒 = 𝑒⊕𝑎 = 𝑎 for all 𝑎 ∈ 𝐆
• Inverse element: For each element 𝑎 ∈ 𝐆, there is an element 𝑏 such that 𝑎⊕𝑏 = 𝑏⊕𝑎 = 𝑒 where 𝑒 is an identity of 𝐆.

Another requirement is that the binary operation ⊕ is closed under 𝐆. It means that 𝑎⊕𝑏 ∈ 𝐆 for all (or any) 𝑎, 𝑏 ∈ 𝐆

Group may seem to be strange, but we use it in everyday life. An example of Group is a set of integer with binary operation called addition. Guess what its identity element and an inverse element of each element.

Another one is a set of all even numbers (including 0) with the addition.

Now, consider a set of all the odd numbers with the addition.

{…, -5, -3, -1, 1, 2, 3, …}

Is this a group? It is not a group since there is no identity element. Now, add the identity (0).

{…, -5, -3, -1, 0, 1, 2, 3, …}

This has all three properties: associativity, an identity element and an inverse element -𝑎 for any element 𝑎. But this is not a group. It is not closed under the addition. 1+3 is not in the set.

There are a lot of groups. A few examples are

• Set of real numbers with the addition
• Set of real numbers except the zero element 0 with the multiplication
• Set of permutations of n elements with the composition

Many things are not a group when they are similar to a group, like a set of integers with the minus operation.