Isomorphisms, Automorphisms, Homomorphisms

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Isomorphisms, automorphisms, homomorphisms

Isomorphisms, automorphisms and homomorphisms are all very similar in their basic concept. An isomorphism is a one-to-one correspondence between two abstract mathematical systems which are structurally, algebraically, identical. The structures might be groups, rings, integral domains, fields, vector spaces, etc. In general these structures consist of a set of elements with one or more operations defined on them, such as addition or multiplication, and represent axiomatically defined mathematical structures. The isomorphic correspondence is so defined that corresponding elements will correspond. If the two structures are groups an isomorphism between them is defined as a correspondence such that if elements x and y of the first group correspond to elements x' and y' respectively of the second group, then the product xy of the first group corresponds to the product x'y' of the second group. This simple condition is adequate to insure that like elements correspond i.e that identity elements correspond, inverse elements correspond, etc. and that the two groups are structurally, algebraically identical. When we say that two mathematical structures are isomorphic with each other we mean that they are identical structurally, algebraically; that corresponding elements correspond and their internal workings and mechanisms are identical. Their differences lie only in superficial things like the names we give the elements and the way we denote the law of combination. So, to repeat: An isomorphism is a one-to-one correspondence between two mathematical structures which are structurally, algebraically, equivalent. Two mathematical structures are said to be isomorphic to each other if it is possible to establish a one-to-one correspondence between them in such a way that it meets the criterion for an isomorphism for that particular kind of structure (the criterion for a group isomorphism, a ring isomorphism, etc.).

An automorphism is defined as an isomorphism of a set with itself. Thus where an isomorphism is a one-to-one mapping between two mathematical structures an automorphism is a one-to-one mapping within a mathematical structure, a mapping of one subgroup upon another, for example.

A homomorphism is also a correspondence between two mathematical structures that are structurally, algebraically identical. However, there is an important difference between a homomorphism and an isomorphism. An isomorphism is a one-to-one mapping of one mathematical structure onto another. A homomorphism is a many-to-one mapping of one structure onto another. Where an isomorphism maps one element into another element, a homomorphism maps a set of elements into a single element. Its definition sounds much the same as that for an isomorphism but allows for the possibility of a many-to-one mapping. The best way to illustrate a homomorphism is in its application to the mapping of quotient groups. Quotient groups are groups whose elements are sets -- namely cosets of the normal group of some group. The cosets of any normal subgroup N of a group G form a group under complex multiplication and this group is called the quotient group (or factor group) of G by N and is denoted by G/N. The normal subgroup N plays the role of the identity in the quotient group. Now let us state a theorem fundamental for the whole theory of homomorphic mappings:

Theorem. Under homomorphic mapping of an arbitrary group G onto a group G', the set N of elements of G that are mapped into the neutral element e' of G' is a normal subgroup of G; the set of elements of G that are mapped into an arbitrary fixed element of G' is a coset of G with respect to N, and the one-to-one correspondence so established between the cosets of G with respect to N and the elements of G' is an isomorphism between G' and the factor group G/N.

theorem excerpted from Mathematics, Its Content, Methods and Meaning. Vol. III, p. 304)

From this theorem we see that under this homomorphism all the elements in a particular coset are imaged into a single element in G'. Elements from separate cosets are imaged into separate elements in G'. Thus we see that under a homomorphic mapping, on transition from G to G', distinct elements of G coalese into a single element of G'. Classes or sets of elements map into single elements.