2
CHAPTER 5. SOME BASIC TECHNIQUES OF GROUP THEORY
(1)
h
(
gx
)=(
hg
)
x
for every
g,h
∈
G
(2) 1
x
=
x
for every
x
∈
X
.
As in (5.1.1),
x
→
deFnes a permutation of
X
. The main point is that the action
of
g
is a permutation because it has an inverse, namely the action of
g
−
1
. (Explicitly, the
inverse of
x
→
is
y
→
g
−
1
y
.) Again as in (5.1.1), the map from
g
to its associated
permutation Φ(
g
) is a homomorphism of
G
into the group
S
X
of permutations of
X
. But
we do not necessarily have an embedding. If
=
hx
for all
x
, then in (5.1.1) we were
able to set
x
= 1, the identity element of
G
, but this resource is not available in general.
We have just seen that a group action induces a homomorphism from
G
to
S
X
, and
there is a converse assertion. If Φ is a homomorphism of
G
to
S
X
, then there is a
corresponding action, deFned by
=Φ(
g
)
x,x
∈
X
. Condition (1) holds because Φ is a
homomorphism, and (2) holds because Φ(1) must be the identity of
S
X
. The kernel of Φ
is known as the
kernel of the action
; it is the set of all
g
∈
G
such that
=
x
for all
x
,
in other words, the set of
g
’s that Fx everything in
X
.
5.1.3 Examples
1. (
The regular action
) Every group acts on itself by multiplication on the left, as
in (5.1.1). In this case, the homomorphism Φ is injective, and we say that the action is
faithful
.
[Similarly, we can deFne an action on the right by (
xg
)
h
=
x
(
gh
),
x
1=
x
, and then
G
acts on itself by right multiplication. The problem is that Φ(
)=Φ
(
h
)
◦
Φ(
g
), an
antihomomorphism. The damage can be repaired by writing function values as
xf
rather
than
f
(
x
), or by deFning the action of
g
to be multiplication on the right by
g
−
1
.W
e
will avoid the diﬃculty by restricting to actions on the left.]
2. (