Mappings between groups

28.5 MAPPINGS BETWEEN GROUPS
28.5 Mappings between groups
Now 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
as XY , 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.
If G and G are two groups, we can study the effect of a mapping
Φ : G → G
of G onto G . If X is an element of G we denote its image in G under the mapping
Φ by X = Φ(X).
A technical term that we have already used is isomorphic. We will now define
it formally. Two groups G = {X, Y , . . . } and G = {X , Y , . . . } are said to be
isomorphic if 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 groups G and G are isomorphic, and G and G are isomorphic,
then it follows that G and G are isomorphic. We have already seen an example of
four groups (of functions of x, 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, the
wider 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.
Let G and G be two groups and Φ a mapping of G → G . If for every pair of
elements X and Y in G
(XY ) = X Y then Φ is called a homomorphism, and G is said to be a homomorphic image of G.
The essential defining relationship, expressed by (XY ) = X Y , is that the
same 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.
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Three immediate consequences of the above definition are proved as follows.
(i) If I is the identity of G then IX = X for all X in G. Consequently
X = (IX) = I X ,
for all X in G . Thus I is the identity in G . In words, the identity element
of G maps into the identity element of G .
(ii) Further,
I = (XX −1 ) = X (X −1 ) .
That is, (X −1 ) = (X )−1 . In words, the image of an inverse is the same
element in G as the inverse of the image.
(iii) If element X in G is of order m, i.e. I = X m , then
· · · X; .
I = (X m ) = (XX m−1 ) = X (X m−1 ) = · · · = X
8 X 9:
m factors
In words, the image of an element has the same order as the element.
What distinguishes an isomorphism from the more general homomorphism are
the requirements that in an isomorphism:
(I) different elements in G must map into different elements in G (whereas in
a homomorphism several elements in G may have the same image in G ),
that is, x = y must imply x = y;
(II) any element in G must be the image of some element in G.
An immediate consequence of (I) and result (iii) for homomorphisms is that
isomorphic groups each have the same number of elements of any given order.
For a general homomorphism, the set of elements of G whose image in G is I is called the kernel of the homomorphism; this is discussed further in the
next section. In an isomorphism the kernel consists of the identity I alone. To
illustrate both this point and the general notion of a homomorphism, consider
a mapping between the additive group of real numbers and the multiplicative
group of complex numbers with unit modulus, U(1). Suppose that the mapping
→ U(1) is
Φ : x → eix ;
then this is a homomorphism since
(x + y) → ei(x+y) = eix eiy = x y .
However, it is not an isomorphism because many (an infinite number) of the
elements of
have the same image in U(1). For example, π, 3π, 5π, . . . in
all
have the image −1 in U(1) and, furthermore, all elements of of the form 2πn,
where n is an integer, map onto the identity element in U(1). The latter set forms
the kernel of the homomorphism.
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