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.