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\title{Math 120 Class Notes}
\author{Lucas Garron}
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\section*{Mar 28, 2011}

{\definition Group.} A group $(G, \circ)$ is a pair of 

\li
\item a set $G$, and
\item a rule that takes in two elements $a, b \in G$ and outputs one element $a \circ b$. (a function $:G \times G \-> G$)
\lid

satisfying the following axioms:

\li
\item for any $x, y, z \in G$, $x \circ (y \circ z) = (x \circ y) \circ z$ (associative law)
\item there is an element $e \in G$ such that $x \circ e = e \circ x = x$ [Note: Unique. Can prove one order from the other.]
\item for every $x \in G$, there is an element $y \in G$ such that $x \circ y = y \circ x = e$ [Note: Unique.]
\lid

{\note $e$ is unique.} If $e, e'$ both have the property of $(2)$, $e' = e \circ e' = e$
{\note Similarly, the $y$ from $(3)$ is unique.} It is called the \emph{inverse} of $x$, denoted by $x\i$ (e.g. $e\i = e$)

{\example Some example groups:}
\li
\item $G =$ all rotations of the sphere.\\
$\circ$: $R \circ R' =$ $R'$ followed by $R$
$e$: rotation that does nothing
\item $G = \R$ (real numbers)\\
$x \circ y = x+y$\\
$e=0$\\
``$x \i$''$= -x$
\item $G = \R \setminus \{0\}$\\
$x \circ y = x\*y$\\
$e = 1$\\
$x\i = 1/x$
\item Permutations. A \emph{permutation} of $X$ is a bijective function $X \-> X$.\\
If $\sigma, \sigma'$ are permutations, then so is $\sigma \circ \sigma' =$ permutation given by first applying $\sigma'$ then $\sigma$. (Think $(\sigma \circ \sigma')(x) = \sigma(\sigma'(x))$).\\
For any set $X$, $($set of permutations of $X$, $\circ$) is a group.
\lid

{\example Take $X = \{1, 2, 3\}$.} There are $6 = 3!$ permutations of $X$.

{\definition Symmetric group of $X$.} $Sym(X) = $ (permutations of $X$, $\circ$)

(Relationships between groups: section $1.6$)

{\definition Homomorphism.} $\varphi:(G \circ)\->(G', \circ')$ is a function $\varphi: G \-> G'$ such that $\varphi(x \circ y) = \varphi(x) \circ' \varphi(y)$.


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