Graphics Reference
In-Depth Information
Definition.
The
kernel
of f, ker(f), and the
image
of f, im(f), are defined by
()
=Œ
()
=
{
1
ker f
g
G f g
and
()
=
{
()
ŒŒ
}
.
im f
f g
H
g
G
B.4.13. Lemma.
(1) The sets ker(f) and im(f) are subgroups of G and H, respectively, with ker(f) a
normal subgroup.
(2) The homomorphism f is one-to-one if and only if ker(f) =
1
.
Proof.
Easy.
Now let H be a subgroup of a group G.
Definition.
Let g ΠG. The sets
{
}
gH
=
gh h
Œ
H
and
{
}
Hg
=
hg h
Œ
H
are called the
left
and
right coset
, respectively, of H in G generated by g. If G is abelian,
then the left and right coset are the same set and will be called simply a
coset
in that
case.
It can be shown that two left cosets of H in G are either the same sets or they are
totally disjoint and the union of all left cosets is G. The same holds for right cosets.
In the abelian case, if we define a relation ~ in G by
g
~
g
if
g
-Œ
g
H
,
1
2
1
2
then ~ is an equivalence relation and the cosets of H in G are nothing but the equiv-
alence classes of ~.
B.4.14. Example.
The cosets of {0,3,6} in
Z
9
are {0,3,6}, {1,4,7}, and {2,5,8} .
The cosets of
Z
=
Z
1
in
Z
2
are the sets {(n,k) | n Œ
Z
} for integers k.
B.4.15. Example.
Cosets have the same number of elements. One can also show that every left coset
is a right coset if the subgroup is normal.
Definition.
The
factor
or
quotient group
of a group G by a normal subgroup H is
defined to be the pair (G/H, ·¢), where
