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Math 3124
Thursday, October 20
Sixth Homework Solutions
1. Problem 17.9. Assume that
G
is a group with a subgroup
H
such that

H

=
6,
[
G
:
H
]
>
4, and

G

<
50. What are the possibilities for

G

?
By Lagrange’s theorem and using the given hypotheses,

G

=
6
[
G
:
H
]
<
50 and
[
G
:
H
]
>
4. Therefore the possibilities for
[
G
:
H
]
are 5,6,7,8 and hence the possibilities
for

G

are 30, 36, 42 and 48.
2. Problem 17.17. Find all the subgroups of
Z
36
. Also construct the subgroup lattice.
Since
Z
36
is a cyclic group of order 36, there will be exactly one subgroup for each
positive integer dividing 36. The integers dividing 36 are 2
a
3
b
where
a
,
b
=
0
,
1
,
2, so
there will be 3
*
3
=
9 subgroups. The subgroup lattice will look like
±
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@
@
@
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@
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@
@
@
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@
@
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@
@
@
@
@
@
u
h
[
0
]
i
u
h
[
12
]
i
u
h
[
18
]
i
u
h
[
6
]
i
u
h
[
2
]
i
u
h
[
9
]
i
u
h
[
4
]
i
u
h
[
3
]
i
u
h
[
1
]
i
3. From
ae
=
c
, we see that neither
a
nor
e
is the identity, and from
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This note was uploaded on 01/02/2012 for the course MATH 3124 taught by Professor Parry during the Fall '08 term at Virginia Tech.
 Fall '08
 PARRY
 Math, Algebra

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