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Math 113, Spring 2011
Professor Mariusz Wodzicki
Final Exam (Solutions)
May 11, 2011
1.
Classify the group
G
=
(
Z
×
Z
)/
⟨
(5,6)
⟩
according to the
Fundamental Theorem of Theory of
Finitely Generated Abelian Groups
.
Let
π
:
Z
×
Z
→
G be the quotient map and
ι
:
⟨
(1,1)
⟩→
Z
×
Z
be the inclusion map. We will show that
φ
=
π
◦
ι
is an isomorphism which means that G is inﬁnite cyclic. Indeed,
⟨
(1,1),(5,6)
⟩
contains both
(1,0)
and
(0,1)
:
(1,0)
=
6
·
(1,1)

(5,6)
and
(0,1)
=
5
·
(1,1)
+
(5,6)
which generate
Z
×
Z
, and thus
⟨
(1,1),(5,6)
⟩ =
Z
×
Z
. It follows that
φ
is surjective. It is obviously injective,
since
⟨
(1,1)
⟩∩⟨
(5,6)
⟩=
{(0,0)}
.
2.
Show that the rings 2
Z
and 3
Z
are not isomorphic.
Any ring isomorphism is an isomorphism of the corresponding additive groups. Both
2
Z
and
3
Z
are inﬁnite
cyclic, and an isomorphism between
2
Z
and
3
Z
must take
2
to a generator of
3
Z
, i.e. to
3
or

3
. In partic
ular,
4
=
2
+
2
would be sent to either
6
=
3
+
3
, or

6
=
3
+
(

3)
instead of
9
=
(
±
3)
2
. Thus, none of the two
isomorphisms of the additive groups is a homomorphism of multiplicative semigroups.
3.
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This note was uploaded on 10/15/2011 for the course CHEM 1 taught by Professor Gelfand during the Summer '11 term at Solano Community College.
 Summer '11
 gelfand
 Chemistry

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