Graphics Reference
In-Depth Information
B.4.6. Example.
For each integer k,
{
}
k nn
Z
=
Œ
Z
is a subgroup of
Z
(with respect to addition) and all subgroups of
Z
are of that form.
Z
n
is a subgroup of
R
n
(with respect to vector addition).
B.4.7. Example.
B.4.8. Example.
The set {0,3,6} defines a subgroup of
Z
9
.
B.4.9. Example.
The set {2n | n Œ
Z
} » {3} is
not
a subgroup of
Z
.
The next lemma gives a simple criterion for when a subset of a group is a
subgroup.
B.4.10. Lemma.
(1) A nonempty subset H of a group G is a subgroup (under the operation induced
from that of G) if and only if the element h
1
h
2
-1
belongs to H for all h
1
and
h
2
in H.
(2) The intersection of an arbitrary number of subgroups is a subgroup.
Proof.
Straightforward.
Definition.
A subgroup H of a group G is said to be a
normal subgroup
of G if for
all g in G and all h in H, ghg
-1
belongs to H.
Note that every subgroup of an abelian group is normal.
Definition.
Let G and H be groups. A map f : G Æ H is called a
homomorphism
if
fgg
(
)
=
fg fg
()()
12
1
2
for all g
1
and g
2
in G. The homomorphism is said to be an
isomorphism
if it is a bijection.
In that case we say that the group G is
isomorphic
to the group H and write G ª H.
It is easy to see that homomorphisms map the identity to the identity and inverses
to inverses. In the abelian case this means that f(0) = 0 and f(-g) =-f(g) for a homo-
morphism f. If f is an isomorphism, then so is its inverse f
-1
.
B.4.11. Example.
Inclusion maps such as
Z
Ã
Q
Ã
R
are clearly homomorphisms.
B.4.12. Example.
Define a map
p
n
:
ZZ
Æ
n
as follows: If k Œ
Z
and k = an + b, where a and b are integers and 0 £ b < n, then let
p
n
(k) = b. The map p
n
is a homomorphism.
Next, let f : G Æ H be a homomorphism of groups.
