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238
5. ACTIONS OF GROUPS
Example 5.1.4.
Any group
G
acts on itself by left multiplication. That
is, for
g
2
G
and
x
2
G
,
gx
is just the usual product of
g
and
x
in
G
. The homomorphism, or associative, property of the action is just the
associative law of
G
. There is only one orbit. The action of
G
on itself by
left multiplication is often called the
left regular action
.
Deﬁnition 5.1.5.
An action of
G
on
X
is called
transitive
if there is only
one orbit. That is, for any two elements
x;x
0
2
X
, there is a
g
2
G
such that
gx
D
x
0
. A subgroup of Sym
.X/
is called
transitive
if it acts
transitively on
X
.
Example 5.1.6.
Let
G
be any group and
H
any subgroup. Then
G
acts
on the set
G=H
of left cosets of
H
in
G
by left multiplication,
g.aH/
D
.ga/H
. The action is transitive.
Example 5.1.7.
Any group
G
acts on itself by conjugation: For
g
2
G
,
deﬁne
c
g
2
Aut
.G/
±
Sym
.G/
by
c
g
.x/
D
gxg
±
1
. It was shown in
Exercise
2.7.6
that the map
g
7!
c
g
is a homomorphism. The orbits of
this action are called the
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 Fall '08
 EVERAGE
 Algebra, Multiplication

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