In our exposition of the field axioms, we noted that a group is a set G equipped with an identity element 1, a function -1 from G to G called ``inverse'', and a function (``group operation'') from GxG to G denoted by *, satisfying the axioms:
i) For all a in G, a*1=1*a=a
ii) For all a in G, a*a-1=a-1*a=1
iii) For all a,b,c in G, a*(b*c)=a*(b*c)
[we'll usually suppress the ``*'', so for instance the associateive law will be written a(bc)=a(bc)]

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'' Sn. If our space is a vector space V, the automorphisms are invertible linear operators on V, which constitute the ``general linear group'' GL(V). [``General'' as opposed to ``special'', which we'll define for finite-dimensional V once we've sufficiently developed the theory of determinants.] If V is finite dimensional, of dimension n, we may choose a basis and thus identify GL(V) with the group GLn(F) of invertible n-by-n matrices with entries in the field F. If V is endowed with the further structure of an inner product, we obtain a subgroup of GL(V) preserving this inner product, called the ``orthogonal'' or ``unitary'' group according as F is R or C respectively. A choice of orthonormal basis identifies these groups with the groups On, Un of orthogonal and unitary n-by-n matrices.

The identification of the group of permutations of a finite set with Sn, or of the group of invertible linear operators on a finite-dimensional vector space with GLn(F), depends on the choice of identification of the space A with a standard n-element set {1,2,3,...,n} or a standard n-dimensional vector space Fn. But any two different identifications are related by some automorphism of A, i.e. by an element h of the group G. If any automorphism is represented by g under one identification, it is represented by h-1gh in the other. This expression h-1gh is known as the conjugate of g by h. So, for instance, the real spectral theorem applied to invertible operators says that an element of GLn(R) is conjugated to a diagonal matrix by an element of the orthogonal group On if and only if it is self-adjoint.

Given any group G and any element h, the map taking any g in G to h-1gh 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-1hg 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 PGLn(F)].

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/nZ 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 Sn to the two-element group {1,-1}, and use it to define for any finite-dimensional vector space V over a field F the determinant, a function from Hom(V,V) to F. We shall show that the determinant of a linear operator on V vanishes if and only if it is not invertible, and that on the invertible operators it yields a homomorphism from GL(V) to F*. The kernel of this homomorphism, consisting of all linear transformations of V whose determinant equals 1, is called the ``special linear group'' SL(V), identified with SLn(F) once a basis is chosen for V.