diff --git a/thesis/Makefile b/thesis/Makefile
index 60e5943..d90be32 100644
--- a/thesis/Makefile
+++ b/thesis/Makefile
@@ -5,6 +5,7 @@ bibtex=bibtex
 chapters = chapters/introduction.tex \
 		   chapters/naive_simulator.tex \
 		   chapters/introduction_qc.tex \
+		   chapters/stabilizer.tex \
 		   chapters/graph_simulator.tex
 
 all: main.pdf
diff --git a/thesis/chapters/graph_simulator.tex b/thesis/chapters/graph_simulator.tex
index e426add..f1cbf37 100644
--- a/thesis/chapters/graph_simulator.tex
+++ b/thesis/chapters/graph_simulator.tex
@@ -7,14 +7,6 @@ graph formalism works and how the simulator handles gates.
 
 %A $n$ qbit graph or stabilizer state is a $+1$ eigenstate of some $ p \in P_n$ where $P_n$ is the Pauli group\cite{andersbriegel2005}.
 
-\begin{definition}
-    \begin{equation}
-        C_n := \{U \in SU(2) | UpU^\dagger \in P_n \forall p \in P_n\}
-    \end{equation}
-    is called the Clifford group on $n$ qbits.
-    $C_1$ is called the local Clifford group.
-\end{definition}
-
 One can show that the Clifford group $C_n$ can be generated using the elements of $C_1$  acting on all qbits and
 the controlled phase gate $CZ$ between all qbits\cite{andersbriegel2005}. It is worth noting that the $CX$ gate can be
 generated using $CZ$ and $C_1$ gates.
diff --git a/thesis/chapters/stabilizer.tex b/thesis/chapters/stabilizer.tex
index a1ed329..1b84384 100644
--- a/thesis/chapters/stabilizer.tex
+++ b/thesis/chapters/stabilizer.tex
@@ -19,7 +19,7 @@ Where $X = \left(\begin{array}{cc} 0 & 1 \\ 1 & 0\end{array}\right)$,
     $Z = \left(\begin{array}{cc} 1 & 0 \\ 0 & -1\end{array}\right)$ are the Pauli matrices and 
     $I$ is the identity. 
 
-    $p \in = P_n$ is called a multi-local Pauli operator.
+    $p \in P_n$ is called a multi-local Pauli operator.
 \end{definition}
 
 \begin{definition}
@@ -41,16 +41,46 @@ Where $X = \left(\begin{array}{cc} 0 & 1 \\ 1 & 0\end{array}\right)$,
 
 \begin{lemma}
     For every $\langle S_i \rangle_i$ fulfilling the first two conditions in definition \ref{def:stabilizer} there exists
-    a (up to a global phase) unique state $\ket{\psi}$ fulfilling the third condition.
+    a (up to a global phase) unique state $\ket{\psi}$ fulfilling the third condition. This state is called stabilizer state.
 \end{lemma}
 
 \begin{proof}
-    All multi-local Pauli operators are hermitian (observables in terms of quantum mechanics), as they
+    All multi-local Pauli operators are hermitian (observables in terms of quantum mechanics), as the $S_i$
     commute they have a common set of eigenstates. Because each $S_i$ has eigenvalues $+1, -1$, there 
     exist $2^n$ eigenstates, one state $\ket{\psi}$ with eigenvalue $+1$ for all $S_i$. As the dimension of $n$ qbits is $2^n$
     the state $\ket{psi}$ is unique up to a global phase.
 \end{proof}
 
+One can study the dynamics of stabilizer states using only the stabilizers\cite{nielsen_chuang_2010}. Two important
+cases are the unitary transformation of a state and the measurement of a qbit. When applying a unitary gate to a stabilizer
+state $\ket{\psi}$ the resulting state will in general be no stabilizer state anymore, however there exists a group of
+transformations that map stabilizers to other stabilizers: the Clifford group.
+
+\begin{definition}
+    \begin{equation}
+        C_n := \{U \in SU(2) | UpU^\dagger \in P_n \forall p \in P_n\}
+    \end{equation}
+    is called the Clifford group on $n$ qbits.
+    $C_1$ is called the local Clifford group.
+\end{definition}
+
+The properties of this group will be discussed later, for the time being is existence is enough. 
+
+\begin{lemma}
+    Let $\ket{\psi}$ be stabilized by $\langle S_i \rangle_i$, then $U\ket{\psi}$ is stabilized
+    by $\langle US_iU^\dagger \rangle_i$.
+\end{lemma}
+
+\begin{proof}
+$$ U\ket{\psi} = US_i\ket{\psi} = US_iU^\dagger U\ket{\psi}$$
+So $U\ket{\psi}$ is a $+1$ eigenstate of $US_iU^\dagger$.
+\end{proof}
+
+This is an important insight that is used for simulations\cite{gottesman_aaronson2008}, as
+updating the $n$ stabilizers that are a tensor product of $n$ Pauli matrices scales with roughly
+$\mathcal{O}(n^2)$ instead of $\mathcal{O}(2^n)$ for the state vector approach.
+
+
 \subsection{The Vertex Operator-Free Graph States}
 
 In order to understand some essential transformations of graph states it is necessary