Let $G$ be a Set together with a function $m:G\times G \rightarrow G$, a function $i:G \rightarrow G$, and a distinguished element $e\in G$. The structure $\mathbb{G} = (G,m,i,e)$ is called a $\textdef{group}$ if the following hold for all $x,y,z \in G$; we write $xy$ in place of $m(x,y)$ and $x^{-1}$ in place of $i(x)$.

1. $(xy)z = x(yz)$
2. $xx^{-1} = x^{-1}x = e$
3. $xe = ex = x$

We usually simply write $G$ when referring to the entire structure $\mathbb{G}$.

• We usually think of $m$ as defining a multiplication on $G$, though often this is better interpreted as addition.
• We usually think of $i$ as defining inversion on $G$, though often this is better interpreted as negation.
• When thinking of $m$ as multiplication we often use the symbol $1$ instead of $e$. Similarly, When thinking of $m$ as addition we may use the symbol $0$ instead of $e$.

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\begin{definition}
Let $G$ be a set together with a function $m:G\times G \rightarrow G$, a function $i:G \rightarrow G$, and a distinguished element $e\in G$. The structure $\mathbb{G} = (G,m,i,e)$ is called a \textdef{group} if the following hold for all $x,y,z \in G$; we write $xy$ in place of $m(x,y)$ and $x^{-1}$ in place of $i(x)$.
\begin{enumerate}
\item $(xy)z = x(yz)$
\item $xx^{-1} = x^{-1}x = e$
\item $xe = ex = x$
\end{enumerate}
We usually simply write $G$ when referring to the entire structure $\mathbb{G}$.
\end{definition}

• Group

definition