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\title{18.701 Lecture Notes}
\renewcommand{\today}{2007-09-14}

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\definition{A \dt{partition} of a set $S$ is a decomposition of $S$
  into nonempty disjoint subsets \\

\[ S = U_1\union U_2\union\ldots\union U_n \]
\[ U_1\isect U_2 = \emptyset \mbox{ for } i \neq j\]
\[ \bar{S} = \{U_1,U_2,\ldots\} \]
So we have a surjective map $\pi:S\to\bar{S}$
}

\definition{Given $f:S\to T$, the \dt{fiber} of some $t\in T$ is the
  inverse image of $t$: $f^{-1}(t) = \{s|f(s)=t\}$}

The nonempty fibers of any function partition $S$.

\definition{An \dt{equivalence relation} on $S$, $a \sim b$, is 
  \begin{itemize}
  \item transitive: $a\sim b \land b\sim c \Rightarrow a\sim c$
  \item symmetric: $a\sim b \Rightarrow b \sim a$
  \item reflexive $a\sim a \forall a \in S$
  \end{itemize}
}

\proposition{The equivalence relationships on $S$ correspond
  bijectively to the partitions on $S$}

Given a homomorphism $\phi:G\to G'$, $a,b$ in the same fiber if
$\phi(a)=\phi(b) \iff \phi(a^{-1}b) = 1 \iff a^{-1}b\in\ker\phi$

Say $\ker\phi = N$. $a^{-1}b\in N \Rightarrow b=an, n\in N$

\definition{A \dt{left coset} of $N$ in $G$: 
\[aN=\left\{x\in G|x=an, n\in N\right\}\]}

Let $H\subset G$ Define the left cosets $aH$. These partition $G$ The
corresponding equivalence relationship: $a\cong b \iff a^{-1}b\in
H \rightarrow b\in aH$

All the cosets have the same order.

\theorem{$|G|=|H|$(\# cosets) $= |H|[G:H]$}
\corollary{$|H|$ divides $|G|$}
\corollary{$|[G:H]|$ divides $|G|$}

\corollary{Suppose $|G|=p$ prime. Every subgroup is either $\{1\}$ or
  $G\Rightarrow$ and $G$ is cyclic}




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