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2.4. HOMOMORPHISMS AND ISOMORPHISMS
119
(c)
Suppose that
˛
and
ˇ
are elements of
S
n
and that
ˇ
D
g˛g
±
1
for
some
g
2
S
n
. Show that when
˛
and
ˇ
are written as a product
of disjoint cycles, they both have exactly the same number of
cycles of each length. (For example, if
˛
2
S
10
is a product of
two 3–cycles, one 2–cycle, and four 1–cycles, then so is
ˇ
.) We
say that
˛
and
ˇ
have the same cycle structure
.
(d)
Conversely, suppose
˛
and
ˇ
are elements of
S
n
and they have
the same cycle structure. Show that there is an element
g
2
S
n
such that
ˇ
D
g˛g
±
1
.
The result of this exercise is as follows: Two elements of
S
n
are conjugate
if, and only if, they have the same cycle structure.
2.4.15.
Show that
±
is the unique homomorphism from
S
n
onto
f
1;
±
1
g
.
Hint:
Let
'
W
S
n
±! f˙
1
g
be a homomorphism. If
'..12//
D ±
1
, show,
using the results of Exercise
2.4.14
, that
'
D
±
. If
'..12//
D C
1
, show
that
'
is the trivial homomorphism,
'.²/
D
1
for all
²
.
2.4.16.
For
m < n
, we can consider
S
m
as a subgroup of
S
n
. Namely,
S
m
is the subgroup of
S
n
that leaves ﬁxed the numbers from
m
C
1
to
n
. The
parity of an element of
S
m
can be computed in two ways: as an element of
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 Fall '08
 EVERAGE
 Algebra

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