Mappings and equivalence

Mappings between groups which conserve their group products are called homomorphisms, with a full range of adjectives to distinguish self-maps, surjective and injective maps. Each mapping defines an equivalence relation, compatible to a greater or lesser extent with the group multiplication. Sometimes the equivalence classes themselves are groups, multiplication having been defined setwise rather than pointwise.

Finally there are different kinds of multiplication which can be defined for cartesian products, still satisfying the requirement that projections are homomorphisms. The study of their existence and properties would complete the structure theory.

Not all equivalence relations in a group correspond to homomorphisms. Those which do are called congruencecongruence relation relations. The two basic equivalence relations, whose equivalence classes are the cosets, are defined with respect to a subgroup, $H$ say.

$$\begin{eqnarray*} x \equiv y & \Leftrightarrow & xy^{-1} \varepsilon H\ \ [{\rm... ...\Leftrightarrow & x^{-1}y \varepsilon H\ \ [{\rm right\ coset}] \end{eqnarray*}$$

In words, two elements are equivalent whenever they are common multiples of elements taken from a subgroup. The idea is that a subgroup should be seen as the identity element by a mapping which is unwilling to distinguish between its members. What we want is that $ax \equiv ay$ whenever $x \equiv y$, say. To get rid of whastever $a$ we should test $(ay)^{-1}(ax)$.

For what should we test? We want $x \equiv x$, so $e$ should be one of the quotients. We need $x \equiv y \Leftrightarrow y \equiv x$, so we should always have $xy^{-1}$ along with $yx^{-1}$, so quotients should always be paired with their inverses. Finally, the transitive law requires products, so all the requirements for a subgroup have been specified without saying which subgroup. Any will do, so there are cosets for all subgroups, left or right according to the handedness of the group multiplication required.

To get a congruence relation, and equivalence irrespective of the factor being replaced, a subgroup is required whose left cosets are the same as its right cosets; such a subgroup is called a normalnormal subgroup subgroup. Its cosets are then congruence classes, the subgroup itself is the counterimage of the identity subgroup with respect to a homomorphism.

Homomorphic images of subgroups are subgroups. So are homomorphic counterimages.

Equivalence relations can be ordered by inclusion of their equivalence classes. Upper and lower bounds of cosets of the same handedness follow from the ordering of their defining subgroups; between handednesses the structures are called doubledouble coset cosets.

There is still another important equivalence relation, whose equivalence classes are simply called classesclass.

$$\begin{eqnarray*} x \equiv y & \Leftrightarrow & \exists a \ni a x = y a. \end{eqnarray*}$$

It is the same relationship. applied internally to a group and its own group multiplication that characterizes equivalence in terms of mappings and commutativecommutative diagram diagrams.

Because group multiplication is always invertible, multiplication by a fixed factor permutes the group elements; that is to say, if $ax = ay$ it follows that $x=y$. For that reason, any group can be regarded as a group of permutations. If the permutations were written as matrices, that would also exhibit the group as a collection of matrices, with matrix multiplication as the group operation. But there is another approach which is capable of characterizing all the possible sets of matrices homomorphic to a given group.