i) For all

ii) For all

iii) For all

[we'll usually suppress the ``*'', so for instance the associateive law will be written

You may have wondered why we have a special term for an operation
satisfying the associative rule rather than the simpler commutative
rule. The reason is that associativity naturally arises for the
set of *symmetries* (a.k.a. *automorphisms*) of
a mathematical space *A*, i.e. of bijections of *A*
that preserve the structure of *A*. When we impose no
structure, the automorphism group is just the group of all
permutations of *A*; if *A* is a finite set of size
*n*, this group is called the ``symmetric group''
*S _{n}*. If our space is a vector space

The identification of the group of permutations of a finite set with
*S _{n}*, or of the group of invertible linear operators
on a finite-dimensional vector space with

Given any group *G* and any element *h*, the map taking
any *g* in *G* to *h*^{-1}*gh*
is an automorphism of *G*. An automorphism of this form
is called an ``inner automorphism''; all such automorphisms
are ``trivial'' (the identity) if and only if *G* is
a commutative group. Such groups may still have nontrivial
automorphisms (such as the automorphism of the additive group
**Z** taking each *a* to *-a*).
An automorphism which is not inner is called ``outer'' (surprise!).
As with vector spaces, groups can be studied via a natural
generalization of automorphisms called ``homomorphisms''.
A homomorphism between two groups *G* and *G'*
is a map *f* that respects the group operations:
*f*(1)=1,
*f*(*a*^{-1})=(*f*(*a*))^{-1}
for all *a* in *G*, and
*f*(*ab*)=*f*(*a*)*f*(*b*)
for all *a,b* in *G*. Again the image and kernel
of a homomorphism are *subgroups* of *G'* and *G*
respectively, i.e. subsets containing the identity and closed under
inverse and *, which thus become groups in their own right.

As with homomorphisms of vector spaces (i.e. linear transformations),
any subgroup *H* of a group *G* is the image of a
homomorphism, namely the inclusion homomorphism from *H*
to *G* obtained by sending each element of *H* to itself.
However, unlike the situation for vector spaces, *not* every
subgroup can be the kernel of a homomorphism. The subgroups of
*G* which *are* kernels of homomorphisms are called
*normal subgroups* of *G*. A subgroup *N* is
normal if and only if it contains the conjugate
*g*^{-1}*hg* of any *h* in *N*
by any *g* in *G* (not only in *N*!). The
``only if'' is easy; for ``if'', construct a *quotient group*
*G/N*, and then consider the natural homomorphism from
*G* to *G/N*. For instance, *GL*(*V*)
contains as a normal subgroup the nonzero scalar multiples of
the identity; the quotient group is called the *projective
general linear group* of *V* and denoted by
*PGL*(*V*) [the matrix version of this is
naturally denoted *PGL _{n}*(

You may have heard or read of *simple* groups. For any
group *G*, the group *G* itself, and the
one-element subgroup {1}, are clearly normal. *G*
is called ``simple'' if these are the only normal subgroups
of *G*. Simple groups play a role in group theory
analogous to that of primes in number theory; indeed one readily sees
that the group **Z**/*n***Z** is
simple if and only if *n* is prime. But there are other,
much more interesting, examples of simple groups. We shall not
pursue this further in Math 55; the topic is addressed in 100-level
algebra courses.

We will construct a homomorphism ``sign'' from the symmetric
group *S _{n}* to the two-element group {1,-1},
and use it to define for any finite-dimensional vector space