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{\small
\ni{Lectures by Robert Geroch}
\ni{{\TeX}ed and supplemented by R. Salgado}
}

\ni{\bf GROUPS}
\bigskip

A {\bf group}\marginpar{\bf group} is a set $G$, together with a mapping
$G\times G \stackrel{\pi}{\rightarrow} G$ (\ie, a rule that assigns,
for any two elements $g$ and $g'$ of $G$, a third element, $\pi(g,g')$),
satisfying the following three conditions:
     \begin{enumerate}
          \item (Associativity) For $g$, $g'$, and $g''$ any three elements
               of $G$, $\pi(\pi(g,g'),g'')=\pi(g,\pi(g',g''))$.
          \item (Existence of Identity) There exists an element $e$ of $G$
               such that, for every $g\in G$, $\pi(e,g)=\pi(g,e)=g$.
          \item (Existence of inverses) For each element $g$ of $G$, there
               exists an element, $g^{-1}$, of $G$ such that
               $\pi(g,g^{-1})=\pi(g^{-1},g)=e$.
     \end{enumerate}
The element $\pi(g,g')$ is often written simply ``$gg'$''.

A group is said to be {\bf abelian}\marginpar{\bf abelian} if, for any
$g,g'\in G, \pi(g,g')=\pi(g',g)$.  For an abelian group, $\pi(g,g')$ is
often written ``$g+g'$''; the identity ``$0$''; the inverse of $g$ ``$-g$''.

Thus, to specify a group one must say what the set $G$ is; say, for any
$g, g' \in G$, what element of $G$ $\pi(g,g')$ is; and check the three
conditions above.

\ni EXAMPLES

The reals under addition (\ie, let $G$ be the set of real numbers, and let
$\pi(a,b)$ be $a+b$, the sum of real numbers) is an abelian group.
     \tech{The first condition follows from the fact that the addition of
          numbers is associative; the second from the choice of the number
          ``$0$'' for the identity; and the third from the choice of a
          number ``$-a$'' for the inverse of $a$.  This group is abelian
          because addition is commutative.}\\
Similarly, the complexes under addition, any vector space under addition,
and the integers under addition are abelian groups.

The invertible operators on a Hilbert space, under composition, are a
group.~\tech{Composition of operators is associative; use ``$I$'' for the
          identity; invertible operators have inverses.}\\
Similiarly, the unitary operators on a Hilbert space, under composition,
are a group.

For $S$ any set, the collection of all one-to-one-and-onto maps from $S$
to itself form a group, called the {\bf permutation group} of $S$.
     \tech{The product is composition of maps; the identity is the
          identity map; one-to-one-and-onto maps have inverses.}\\
This group is abelian if and only if $S$ has fewer than three elements.

For fixed integer $n\geq 1$, the set of all invertible, real, $n\times n$
matrices, under matrix multiplication, is a group.  Similarly, the set
of unit-determinant matrices, and the set of orthogonal matrices, are groups.
     \tech{This follows, \eg, in the case of the latter, from the fact
          that the product of two orthogonal matrices is an orthogonal
          matrix.}\\
Similarly, the complex invertible $n\times n$ matrices, and the unitary
matrices, form groups.
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