HOMEWORK 8: GRADER’S NOTES AND SELECTED SOLUTIONS
Grader’s Notes:
●
In general when you are trying to show that two groups
G
and
H
are
not
isomorphic, it’s never enough
to pick one particular map and show it’s not an isomorphism. For example, in showing that the rational
numbers
under addition are not isomorphic to any proper subgroup you can’t just show that the
map
φ
(
x
)
=
x
2
is not an isomorphism because there could, conceivably be some other map that
is
an
isomorphism.
●
The proper way to show that
G
≅
H
in general is usually either:
(1) Find some property that you can prove / have proved is preserved under isomorphism and show
one of
G,H
have this property while the other group does not.
(2) Assume that
φ
∶
G
→
H
is an isomorphism (and assume
nothing more
about
φ
) and somehow arrive
at a contradiction.
●
The properties “onetoone”, “onto”, and “operation preserving” are properties that functions have, it
is impossible for groups to have these properties, when groups
G,H
are isomorphic the property that
holds of the groups is that there exists a
function
f
∶
G
→
H
such that
f
,
the function
is onetoone,
onto, and order preserving.
Chapter 6, page 135, no 28
Prove the quaternion group is not isomorphic to the dihedral group
D
4
.
Example Solution:
Use Theorem 6.2.7 and note that the Quaternions have 1 element of order 2 while
D
4
has 5 elements of order
2. There are variants on this method of proof all related to the fact that if
φ
is an isomorphism then
x
=
φ
(
x
)
.
Grader’s Notes:
●
Note that when two groups
G,H
are given by Cayley tables, an isomorphism
φ
∶
G
→
H
need not take
the
n
×
m
th entry of the Cayley table for
G
to the
n
×
m
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 Spring '08
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 1 m, 1 m, 1 g, Isomorphism, Group isomorphism

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