Math 113 Section 5
Homework #7
Fall 2010
Due: Monday, October 25
1.
Section 4.3, Exercise 5: Let
G
be a group. If the center of
G
is of index
n
, prove that
every conjugacy class has at most
n
elements.
2.
Section 5.1, Exercise 1: Let
G
1
, G
2
, . . . , G
n
be groups. Show that the center of a direct
product is the direct product of the centers:
Z
(
G
1
×
G
2
× · · · ×
G
n
) =
Z
(
G
1
)
×
Z
(
G
2
)
× · · · ×
Z
(
G
n
)
.
Deduce that a direct product of groups is abelian if and only if each of the factors is
abelian.
3.
Section 5.1, Exercise 2: Let
G
1
, G
2
, . . . , G
n
be groups and let
G
=
G
1
× · · · ×
G
n
. Let
I
be a proper, nonempty subset of
{
1
, . . . , n
}
and ket
J
=
{
1
, . . . , n
} 
I
. Define
G
I
to
be the set of elements of
G
that have the identity of
G
j
in position
j
for all
j
∈
J
.
(a)
Prove that
G
I
is isomorphic to the direct product of the groups
G
i
, i
∈
I
.
(b)
Prove that
G
I
is a normal subgroup of
G
and
G/G
I
is isomorphic to
G
J
.
(c)
Prove that
G
is isomorphic to
G
I
×
G
J
.
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 Fall '08
 OGUS
 Math, Algebra, Group Theory, Ring, direct product

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