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Unformatted text preview: 2.4. HOMOMORPHISMS AND ISOMORPHISMS 111 Deﬁnition 2.4.1. A map between groups ' W G ! H is called a homomorphism if it preserves group multiplication, '.g1 g2 / D '.g1 /'.g1 / for
all g1 ; g2 2 G . An endomorphism of G is a homomorphism ' W G ! G .
There is no requirement here that ' be either injective or surjective.
Example 2.4.2. We consider some homomorphisms of the symmetry group
of the square into permutation groups. Place the square card in the .x y /–
plane so that the axes of symmetry for the rotations a, b , and r coincide
with the x –, y –, and z –axes, respectively. Each symmetry of the card induces a bijective map of the space S D f.x; y; 0/ W jx j Ä 1; jy j Ä 1g
occupied by the card. For example, the symmetry a induces the map
4y 5 7! 4 y 5 :
The map associated to each symmetry sends the set V of four vertices of
S onto itself. So for each symmetry of the square, we get an element
. / of Sym.V /. Composition of symmetries corresponds to composition
of maps of S and of V , so the assignment 7! . / is a homomorphism
from the symmetry group of the square to Sym.V /. This homomorphism
is injective, since a symmetry of the square is entirely determined by what
it does to the vertices, but it cannot be surjective, since the square has only
eight symmetries while jSym.V /j D 24.
Example 2.4.3. To make these observations more concrete and computationally useful, we number the vertices of S . It should be emphasized
that we are not numbering the corners of the card, which move along with
the card, but rather the locations of these corners, which stay put. See
3 2 Figure 2.4.1. Labeling the vertices of the square. Numbering the vertices gives us a homomorphism ' from the group
of symmetries of the square into S4 . Observe, for example, that '.r/ D ...
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