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Math 150a: Modern Algebra
Homework 10
This problem set is due Friday, December 7. Do problems 5.6.3, 5.7.3 (using the counting formula and the
stabilizer of a face), 5.7.5, 6.1.6, 6.2.4, 6.2.7, 6.3.2 (left multiplication only), 6.4.2, 6.4.5, and 6.4.12, in
addition to the following:
GK1.
Let
G
be the subgroup of
S
7
generated by
a
= (
1 2 3 4 5 6 7
)
and
b
= (
2 3 5
)(
4 7 6
)
. As part (a) will
show,
G
is a dihedralish group in which the reflection
f
is replaced by an element of order 3.
a.
Show that
bab
−
1
=
a
2
. (This is using my multiplication convention in
S
n
; otherwise you get
bab
−
1
=
a
4
.) Using this, show that
A
=
a
a
A
is a normal subgroup of
G
and that it is complemented
by
B
=
a
b
A
. How many elements are in
G
? Is
B
normal?
b.
Show that
A
is normal and Sylow and show that
B
is Sylow. How many conjugates does
B
have?
(To do this problem it helps enormously to make a standard form for elements of
G
using the
fact that
G
=
AB
.)
GK2.
Let
p
be prime and let
C
p
×
C
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 Spring '03
 Kuperberg
 Algebra, Multiplication, Counting

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