28.5 MAPPINGS BETWEEN GROUPS
28.5 Mappings between groupsNow that we have available a range of groups that can be used as examples,
we return to the study of more general group properties. From here on, when
there is no ambiguity we will write the product of two elements,X•Y, simply
asXY, omitting the explicit combination symbol. We will also continue to use
‘multiplication’ as a loose generic name for the combination process between
elements of a group.
IfGandG′are two groups, we can study the effect of amappingΦ:G→G′ofGontoG′.IfXis an element ofGwe denote itsimageinG′under the mapping
ΦbyX′=Φ(X).
A technical term that we have already used isisomorphic. We will now defineit formally. Two groupsG={X,Y,...}andG′={X′,Y′,...}are said to be
isomorphicif there is a one-to-one correspondence
X↔X′,Y↔Y′,···between their elements such that
XY=Z implies X′Y′=Z′and vice versa.
In other words, isomorphic groups have the same (multiplication) structure,although they may differ in the nature of their elements, combination law and
notation. Clearly if groupsGandG′are isomorphic, andGandG′′are isomorphic,
then it follows thatG′andG′′are isomorphic. We have already seen an example of
four groups (of functions ofx, of orthogonal matrices, of permutations and of the
symmetries of an equilateral triangle) that are isomorphic, all having table 28.8
as their multiplication table.
Although our main interest is in isomorphic relationships between groups, thewider question of mappings of one set of elements onto another is of some
importance, and we start with the more general notion of a homomorphism.
LetGandG′be two groups andΦa mapping ofG→G′. If for every pair of
elementsXandYinG
(XY)′=X′Y′thenΦis called a homomorphism, andG′is said to be a homomorphic image ofG.
The essential defining relationship, expressed by (XY)′=X′Y′, is that thesame result is obtained whether the product of two elements is formed first and
the image then taken or the images are taken first and the product then formed.