A group is a set G with an operation \(\odot\) which satisfies four requirements:

- closure
$$\forall_{a, b \in G}: \; a \odot b \in G$$

- associativity
$$\forall_{a, b, c \in G}: (a \odot b) \odot c = a \odot ( b \odot c)$$

- identity element
$$\exists_{ e \in G} \forall_{ a \in G}: \; a \odot e = e \odot a = a$$

- inverse element
$$\forall_{ a \in G} \exists_{a^{-1} \in G}: \; a \odot a^{-1} = a^{-1} \odot a = e$$

Abelian group satisfies
$$\forall_{a, b \in G}: \; a \odot b = b \odot a$$