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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
23
23
x
x
4y 5 7! 4 y 5 :
0
0
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
Figure 2.4.1.
4
1
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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This note was uploaded on 12/15/2011 for the course MAC 1105 taught by Professor Everage during the Fall '08 term at FSU.
 Fall '08
 EVERAGE
 Algebra, Multiplication

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