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# Homomorphisms and Isomorphisms

DOI link for Homomorphisms and Isomorphisms

Homomorphisms and Isomorphisms book

# Homomorphisms and Isomorphisms

DOI link for Homomorphisms and Isomorphisms

Homomorphisms and Isomorphisms book

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## ABSTRACT

Consider the function which assigns to every real number a E R its absolute value I a I in the set le of non-negative real numbers. This function h : Ill —> R+ has the property that it is compatible with the multiplicative structure of IR, because I a • b 1=lal•IbI• We get the same result whether we multiply the numbers first and then use our mapping, or permute these actions, since calculation with the images proceeds in the same way as calculations with the originals. A corresponding observation can be made about assigning to a square matrix A its determinant I A I, or to a permutation s on the set {1, ... , n} its sign. In these cases the compatibility of the mapping with the operation can be described by the equations

IA•BI = IAI•1B1 and sgn(si o s2) = sgnsi • sgns2. In each of these examples, we have a mapping between the carrier sets of two algebras of the same type, which is compatible with the operations of the algebras. If the mapping between the carrier sets of the two algebras is also a bijection, then the difference between the two algebras amounts only to a relabelling of the elements. This can be seen for instance in the following example. We consider the algebras

A3 = ({(1), (123), (132)1; o) and 23 = ({[O]3, [1]3, [2]3 }; +),

the group of all even permutations (the alternating group) of order three and the cyclic group of order three, respectively. Both algebras have type

(different order)

We begin with the definitions of homomorphism and isomorphism.