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MTHSC 851 (Abstract Algebra)
Dr. Matthew Macauley
HW 6
Due Tuesday March 3nd, 2009
(1) (a) Prove that
∑
i
∈
I
A
i
is a coproduct in the category of abelian groups. Speciﬁcally, let
{
A
i

i
∈
I
}
be a family of abelian groups, and let
ι
i
be the canonical injections, for
i
∈
I
. If
B
is an abelian group and
{
f
i
:
A
i
→
B

i
∈
I
}
a family of homomorphisms,
prove there is a unique homomorphism
f
:
∑
i
∈
I
A
i
→
B
such that
fι
i
=
f
i
for all
i
∈
I
, and this determines
∑
i
∈
I
A
i
uniquely up to isomorphism.
(b) Give an example of how the direct product
Q
i
∈
I
A
i
fails to be a coproduct in the
category of abelian groups.
(2) Prove that the free product
Y
i
∈
I
*
G
i
is a coproduct in the category of groups.
(3) Let
A
1
,
A
2
,
A
be objects in a category
C
, and let
f
i
∈
Hom(
A,A
i
) for
i
= 1
,
2. Suppose
that
B
A
1
g
1
o
A
2
g
2
O
A
f
1
O
f
2
o
and
B
0
A
1
g
0
1
o
A
2
g
0
2
O
A
f
1
O
f
2
o
are pushouts for (
A,A
1
,A
2
,f
1
,f
2
). Prove that
B
and
B
0
are equivalent.
(4) Give an example of a group that is solvable but not nilpotent.
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This note was uploaded on 03/11/2012 for the course MTHSC 851 taught by Professor Staff during the Fall '08 term at Clemson.
 Fall '08
 Staff
 Algebra

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