Math 5125
Friday, August 26
August 26, Ungraded Homework
Exercise 4.1.4 on page 116
Let
S
3
act on the set
Ω
of ordered pairs:
{
(
i
,
j
)

1
≤
i
,
j
≤
3
}
by
σ
((
i
,
j
)) = (
σ
(
i
)
,
σ
(
j
))
. Find the orbits of
S
3
on
Ω
. For each
σ
∈
S
3
ﬁnd the cycle
decomposition of
σ
under this action (i.e., ﬁnd its cycle decomposition when
σ
is considered
as an element of
S
9
– ﬁrst ﬁx a labelling of these nine ordered pairs). For each orbit
O
of
S
3
acting on these nine points pick some
a
∈
O
and ﬁnd the stabilizer of
a
in
S
3
.
There are two orbits under
S
3
, namely
{
(1,1), (2,2), (3,3)
}
and
{
(1,2), (2,3), (3,1), (1,3),
(2,1), (3,2)
}
. The stabilizer of (1,1) in
S
3
is (2 3), and the stabilizer of (1,2) in
S
3
is 1.
The cycle decompositions are given as follows.
(1)
(
(
1
,
1
)
)
(12)
(
(
1
,
1
)(
2
,
2
)
)(
(
1
,
2
)(
2
,
1
)
)(
(
1
,
3
)(
2
,
3
)
)(
(
2
,
3
)(
1
,
3
)
)
(23)
(
(
2
,
2
)(
3
,
3
)
)(
(
2
,
3
)(
3
,
2
)
)(
(
2
,
1
)(
3
,
1
)
)(
(
3
,
1
)(
2
,
1
)
)
(13)
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 Fall '07
 PALinnell
 Math, Algebra, Orbits, The Elements, Disjoint union

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