81
2.140
Let
X
be a disk divided into
n
congruent circular sectors, and let
ρ
be
a rotation by
(
360
/
n
)
◦
.D
e
f
ne a
roulette wheel
to be a
(
q
,
G
)
coloring,
where the cyclic group
G
=h
ρ
i
of order
n
is acting. Prove that if
n
=
6,
then there are
1
6
(
2
q
+
2
q
2
+
q
3
+
q
6
)
roulette wheels having 6 sectors.
Solution.
The group here is
G
=h
ρ
i
of order 6 acting by rotations. Thus,
ρ
=
(
123456
),
ρ
2
=
(
135
)(
246
),
ρ
3
=
(
14
)(
25
)(
36
),
ρ
4
=
(
153
)(
24
),
ρ
5
=
(
165432
).
Hence
P
G
(
x
1
,...,
x
6
)
=
1
6
(
x
6
1
+
2
x
6
+
2
x
2
3
+
x
3
2
)
and
P
G
(
q
,...,
q
)
=
1
6
(
q
6
+
2
q
+
2
q
2
+
q
3
).
2.141
Let
X
be the vertices of a regular
n
gon, and let the dihedral group
G
=
D
2
n
act (as the usual group of symmetries). De
f
ne a
bracelet
to be a
(
q
,
G
)
coloring of a regular
n
gon, and call each of its vertices a
bead
.
(i)
How many bracelets are there having 5 beads, each of which can
be colored any one of
q
available colors?
Solution.
Proceed for the pentagon as we did for the square in
Example 2.139. If
{
v
0
,v
1
,v
2
,v
3
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 Fall '11
 KeithCornell
 Symmetry group, Regular polygon, V3 V4, v2 v4

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