ur mum

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	\begin{center}
		\huge{\textbf{algebraic structures (i)}}\\
	\end{center}
	\begin{multicols}{2}

		% Section 1
		\begin{framed}
			\begin{center}
				\textbf{1 - groups}
			\end{center}
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		% Section 2
		\begin{framed}
			\begin{center}
				\textbf{2 - subgroups}
			\end{center}

			\noindent
			\textbf{\underline{def 2.1}}
			If $(G,*)$ is group and $H (\ne \emptyset) \subseteq G$ and $H$ is \textbf{closed} under $*$, then $H$ is a \textbf{subgroup} of $G$ IFF $H$ is a group itself under $*$ and $H \leq G$.

			\noindent
			\textbf{\underline{thm 2.2}}
			Suppose $(G,*)$ is a group and $H \subseteq G$. Then $H \leq G$ provided that

			(i) $H \ne \emptyset$

			(ii) $\forall x,y \in H, x*y\in H$

			(iii) $\forall x \in H, x^{-1} \in H $(where $x^{-1}$ is inverse from $G$)

			\noindent
			\textbf{\underline{thm 2.3}}
			Suppose $G$ is a group and $H \leq G$ and $K \leq G$. Then $H \cap K \leq G$.\\

			\noindent
			\textbf{\underline{lem 2.4}}
			Let $G$ be a group, $g \in G$. Then $\langle g \rangle \leq G$.\\

			\noindent
			\textbf{\underline{def 2.5}}
			Let $G$ be a group, $g \in G$. The smallest natural number $m$ such that $g^m = e$ is called the \textbf{order} of g.
			If no such natural number exists, we say g has \textbf{infinite order}.

			\noindent
			\textbf{\underline{lem 2.6}}
			Suppose $G$ is a group and $g \in G$ and $g$ has \textbf{infinite} order. Then for $n,m \in \mathbb{Z}$ with $n \neq m$, $g^n \neq g^m$.

			\noindent
			\textbf{\underline{col 2.7}}
			Suppose $G$ is a group and $g \in G$, $g$ having infinite order. Then $\langle g \rangle$ is an infinite subgroup of $G$.

			\noindent
			\textbf{\underline{lem 2.8}}
			Let $g \in G$ and has \textbf{finite} order $n$. Then:

			(i) The elements $g,g^2,...,g^{n-1},g^n = e$ are all distinct

			(ii) Suppose $s,r \in \mathbb{Z}$ with $s=kn+r$, $k\in \mathbb{Z}$ and $0 \leq r \le n$. Then $g^s=g^r$.

			(iii) If $s,r \in \mathbb{Z}$, then $g^s=g^r \iff s \equiv r \mod n$.

			(iv) $g^s=e \iff n|s$.

			\noindent
			\textbf{\underline{col 2.9}}
			If $G$ is a group and $g \in G$ with $g$ having order $n$. Then the order of $\langle g \rangle = \{g,g^2,...,g^{n-1},e\}$ is $n$.\\

			\noindent
			\textbf{\underline{def 2.10}}
			Suppose $G$ is a group, $g \in G$. The centraliser of $g$ is defined as $C(g) = \{x \in g | xg=gx\}$

			\noindent
			\textbf{\underline{lem 2.11}}
			Suppose $G$ is a group, $g \in G$. Then $C(g) \leq G$.

			\noindent
			\textbf{\underline{def 2.12}}
			Let $G$ be any group. Then the \textbf{centre} of $G$ is

			$Z(G)=\{x\in G | xg=gx, \forall g\in G \}$

			\noindent
			\textbf{\underline{lem 2.13}}
			Let $G$ be any group. Then $Z(G) \leq G$.\\

			\noindent
			\textbf{\underline{remarks and friends}}
			\begin{enumerate}
				\item If $H\leq G$, then $e\in H$ (where $e$ is the \textbf{identity element} of G).

				\item If $H,K\leq G$, then $H\cup K$ is not necessarily a subgroup of $G$.

				\item Suppose $G$ a group. Then $e$ is the \textbf{only} element of $G$ to have order 1.

				\item $Scal(n,\mathbb{R}) \leq Diag(n, \mathbb{R}) \leq Tr(n, \mathbb{R}) \leq GL_n(\mathbb{R})$, and $UT(n,\mathbb{R}) \leq GL_n(\mathbb{R})$

				\item We can have similar subgroups of $GL_n(\mathbb{Q}), GL_n(\mathbb{C})$, and $GL_n(\mathbb{Z}_{p})$, $p$ prime.

				\item $g \in C(g)$

				\item In any group $G$, $C(e)=G$

				\item $\langle g \rangle \subseteq C(g)$

				\item $G$ is abelian $\iff C(g)=G, \forall g \in G$

			\end{enumerate}

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		% Section 3
		\begin{framed}
			\begin{center}
				\textbf{3 - cyclic groups}
			\end{center}
		\end{framed}

		% Section 4
		\begin{framed}
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				\textbf{4 - cosets and lagrange's theorem}
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		% Section 5
		\begin{framed}
			\begin{center}
				\textbf{5 - homomorphisms and isomorphisms}
			\end{center}

			\noindent
			\textbf{\underline{def 5.1}}
			A \textbf{homomorphism} from a group $(G,*)$ to a group $(H, \circ)$ is a map $\varphi:G\to H$ such that $\forall a,b\in G$

			$\varphi(a*b) = \varphi(a)\circ\varphi(b)$

			\noindent
			\textbf{\underline{lem 5.2}}
			Suppose G and H are groups and $\varphi:G\to H$ is a homomorphism. Then:

			(i) $\varphi(e_G)=e_H$

			(ii) $\forall g \in G$, $\varphi(g^{-1}) = \varphi(g)^{-1}$

			(iii) $\text{Im}\varphi=\{\varphi(g) | g \in G \} \leq H$\\

			\noindent
			\textbf{\underline{def 5.3}}
			A homomorphism $\varphi:G\to H$ is an \textbf{isomorphism} if $\varphi$ is both $\bm{1-1}$ and \textbf{onto}.\\

			\noindent
			\textbf{\underline{group theoretic properties}}\\
			Let $P$ be a property of groups. P is a \textbf{group theoretic property} if $G$ has $P$ and if $G\cong H$, then $H$ has $P$ too.
			\begin{enumerate}
				\item Being finite
				\item Being infinite
				\item Being abelian
				\item Being cyclic
				\item Having an element of order $n$
				\item Having a non-trivial centre
				\item Having a trivial centre
				\item Having an element of infinite order
			\end{enumerate}


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		% Section 6
		\begin{framed}
			\begin{center}
				\textbf{6 - conjugacy}
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		% Section 7
		\begin{framed}
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				\textbf{7 - normal subgroups}
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		% Section 8
		\begin{framed}
			\begin{center}
				\textbf{8 - factor groups and first isomorphism theorem}
			\end{center}
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		% Repertoise
		\begin{framed}
			\begin{center}
				\textbf{some subgroups of $\bm{GL_n(\mathbb{R})}$ }
			\end{center}
		\end{framed}

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