Math 4124
Wednesday, February 8
Third Homework Solutions
1. 1.5.2 on page 36. Write out the group table for
Q
8
.
1

1
i

i
j

j
k

k
1
1

1
i

i
j

j
k

k

1

1
1

i
i

j
j

k
k
i
i

i

1
1
k

k

j
j

i

i
i
1

1

k
k

i
i
j
j

j

k
k

1
1
i

i

j

j
j
k

k
1

1

i
i
k
k

k
j

j

i
i

1
1

k

k
k

j
j
i

i
1

1
2. 1.6.7 on page 40. Prove that
D
8
and
Q
8
are not isomorphic.
The group
D
8
has one element of order 1, two elements of order 4, and ﬁve elements
of order 2. In fact if we write
D
8
=
{
r
,
s

r
4
=
s
2
=
e
,
rs
=
sr

1
}
, so that the elements
of
D
8
are
r
i
and
sr
i
where
i
=
0
,
1
,
2
,
3, then
r
0
has order 1,
r
and
r
3
have order 4, and
sr
i
(for
i
=
0
,
1
,
2
,
3) and
r
2
have order 2. On the other hand
Q
8
has one element of
order 1, one element of order 2 (namely

1), and six elements of order 4. If
D
8
and
Q
8
were isomorphic, then they would have the same number of elements of order 2
(and also the same number of elements of order 4); since this is not the case, we see
that
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 Spring '08
 Staff
 Math, Algebra, −i

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