52
Now

X

is even, for it is the disjoint union of twopoint subsets consisting
of an element and its inverse. Since

G

is even, we have

Y

even. But
1
∈
Y
, and so there is at least one element
g
∈
Y
with
g
=
1 and
g
2
=
1
(indeed, there are an odd number of such elements.)
2.47
What is the largest order of an element in
S
n
, where
n
=
1
,
2
, . . . ,
10?
Solution.
Denote the largest order of an element in
S
n
by
µ(
n
)
. There is
n
1
2
3
4
5
6
7
8
9
10
µ(
n
)
1
2
3
4
6
6
12
15
20
24
no known formula for
µ(
n
)
, though its asymptotic behavior is known, by a
theorem of E. Landau.
2.48
The
stochastic group
(
2
,
R
)
consists of all those matrices in GL
(
2
,
R
)
whose column sums are 1; that is,
(
2
,
R
)
consists of all the nonsingular
matrices
a c
b d
with
a
+
b
=
1
=
c
+
d
.
Prove that the product of two stochastic matrices is again stochastic, and
that the inverse of a stochastic matrix is stochastic.
Solution.
If
A
and
B
are stochastic matrices, then
AB
=
a
c
b
d
w
y
x
z
=
a
w
+
cx
ay
+
cz
b
w
+
dx
by
+
dz
.
Now the sum of the entries in the
fi
rst column of
AB
is
(
a
w
+
cx
)
+
(
b
w
+
dx
)
=
(
a
+
b
)w
+
(
c
+
d
)
x
=
w
+
x
=
1
.
Similarly, the sum of the entries in the second column of
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 Fall '11
 KeithCornell
 Disjoint union, stochastic matrices

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