Group

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A group $G$ is a set $S$ along with a binary operator $*$ which satisfies certain properties:

Closure: $\forall a \in S, \forall b \in S, a * b \in S$.
Associativity: $\forall a,b,c \in S, (a*b)*c = a*(b*c)$.
Existence of Identity: $\exists e \in S, \forall a \in S, a*e = e*a = a$. Such an $e$ is called the identity of $G$ (denoted as $\operatorname{id}(G)$). It can be proven that the identity is unique.
Existence of Inverses: $\forall a \in S, (\exists l \in S, l*a = e) \wedge (\exists r \in S, a*r = e)$. It can be proven that $l$ is unique, $r$ is unique and $l = r$.

The order of $G$, denoted as $|G|$, is the number of elements in $G$.

For $g \in G$ and $k \in \mathbb{Z}$, \[ g^k = \begin{cases} g*g*\ldots*g \; (k \textrm{ times}) & \textrm{if } k > 0 \\ e & \textrm{if } k = 0 \\ g^{-1}*g^{-1}*\ldots*g^{-1} \; (-k \textrm{ times}) & \textrm{if } k < 0 \end{cases} \]

$\operatorname{order}(g)$ is the smallest positive integer $k$ such that $g^k = e$. If such a $k$ does not exist, $\operatorname{order}(g) = \infty$.

$G$ is said to be commutative or abelian iff $\forall a, b \in G, a*b = b*a$.

The following are very common examples of abelian groups (which is trivial to prove):

Empty group: has only 1 element, which is identity.
$(\mathbb{Z}, +)$, $(\mathbb{Q}, +)$, $(\mathbb{R}, +)$, $(\mathbb{C}, +)$.
$(\mathbb{Q}-\{0\}, \times)$, $(\mathbb{R}-\{0\}, \times)$, $(\mathbb{C}-\{0\}, \times)$.

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Group

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