Graphics Reference
In-Depth Information
j :
Z
Æ G
by
()
=
j k g
.
If ker(j) =
0,
then G ª
Z
; otherwise,
ker
()
=
{
}
kn
k
Œ
Z
for some n Œ
Z
and G ª
Z
n
.
Definition.
Let G be a group and let g be an element of G. Define the
order
of g, o(g),
to be the smallest of the integers k > 0, such that g
k
= 1, if such integers exist; other-
wise, define the
order
of g to be •.
B.4.21. Example.
Let G =
Z
6
= {0,1,2,3,4,5}. Then o(1) = 6 = o(5), o(2) = 3 = o(4),
o(3) = 2, and o(0) = 1.
Definition.
The number of elements in a group G is called the
order
of G and is
denoted by o(G). If G has only a finite number of elements, then G is called a
group
of finite order
, or simply a
finite group
.
Note that two finite groups of the same order need not be isomorphic.
B.4.22. Theorem.
(Lagrange) Let G be a finite group.
(1) If H is a subgroup of G, then o(H) | o(G). In fact, if n is the number of cosets
of H in G, then o(G) = n o(H). The number n is called the
index
of H in G.
(2) If g ΠG, then o(g) | o(G).
Proof.
Part (1) follows from the fact that G is a union of disjoint cosets of H, each
of which has the same number of elements. To prove (2), one only has to observe that
o(g) = o(
Z
g) and apply (1).
Definition.
A
commutator
of a group G is any element in G of the form ghg
-1
h
-1
,
where g, h ΠG.
B.4.23. Theorem.
The set of finite products of commutators of G is a normal sub-
group H called the
derived
or
commutator subgroup
of G. The factor group G/H is
abelian and called the
abelianization
of G.
Proof.
See [Dean66].
Finally, we want to define a free group. Basically, a free group is a group in which
no “nontrivial” identities hold between elements. An example of a trivial identity is
gg
-1
= 1, which holds for all elements g in a group. A nontrivial identity would be
something like g
1
g
2
5
g
3
= 1, if that were to hold for all elements g
i
. Although the idea
