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76
Solution.
The following permutations commute with
(
12
)(
345
)
:
(
1
)(
67
)
(
)(
)(
)(
)
(
)(
)
(
)(
)(
)(
)
(
)(
354
)(
)(
)
(
)(
)
Since there are only 12 permutations commuting with
α
, this must
be all of them.
2.125
(i)
Show that there are two conjugacy classes of 5cycles in
A
5
, each
of which has 12 elements.
Solution.
The hint shows that

C
S
5
(α)
=
5. Since
h
α
i =
5 and
h
α
i≤
C
S
5
(α)
,wehave
h
α
i=
C
S
5
(α)
.By(
i)
,
C
A
5
(α)
=
A
5
∩
C
S
5
(α)
=
A
5
∩h
α
i=h
α
i
,
so that

C
A
5
(α)
5. Therefore, the number of conjugates of
α
in
A
5
is 60
/

C
A
5
(α)
60
/
5
=
12.
(ii)
Prove that the conjugacy classes in
A
5
have sizes 1, 12, 12, 15,
and 20.
Solution.
There are exactly 4 cycle structures in
A
5
:
(
1
)
;
(
123
)
;
(
12345
)
;
(
)(
34
)
. Using Example 2.30, these determine
conjugacy classes in
S
5
of sizes 1, 20, 24, and 15, respectively. In
part (ii), we saw that the class of 5cycles splits, in
A
5
, into two
conjugacy classes of size 12. The centralizer
C
S
5
(
)
consists
of
(
1
), (
), (
132
), (
45
), (
)(
), (
)(
)
;
Only the
f
rst 3 of these are even, and so
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This note was uploaded on 12/21/2011 for the course MAS 4301 taught by Professor Keithcornell during the Fall '11 term at UNF.
 Fall '11
 KeithCornell

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