240
5. ACTIONS OF GROUPS
Definition 5.1.15.
Consider the action of a group
G
on its subgroups by
conjugation. The stabilizer of a subgroup
H
is called the
normalizer
of
H
in
G
and denoted
N
G
.H/
.
According to Corollary
5.1.14
, if
G
is finite, then the number of dis
tinct subgroups
xHx
1
for
x
2
G
is
OEG
W
N
G
.H/Ł
D
j
G
j
j
N
G
.H/
j
:
Since (clearly)
N
G
.H/
H
, the number of such subgroups is no more
than
OEG
W
HŁ
.
Definition 5.1.16.
Consider the action of a group
G
on itself by conjuga
tion. The stabilizer of an element
g
2
G
is called the
centralizer
of
g
in
G
and denoted Cent
.g/
, or when it is necessary to specify the group by
Cent
G
.x/
.
Again, according to the corollary the size of the conjugacy class of
g
,
that is, of the orbit of
g
under conjugacy, is
OEG
W
Cent
.g/Ł
D
j
G
j
j
Cent
.g/
j
:
Example 5.1.17.
What is the size of each conjugacy class in the symmetric
group
S
4
?
Recall that two elements of a symmetric group
S
n
are conjugate in
S
n
precisely if they have the same cycle structure (i.e., if when written as a
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 Fall '08
 EVERAGE
 Algebra, Group Theory, Symmetric group, cycle structure, NG .H

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