ASSIGNMENT 9  MATH235, FALL 2007
Submit by 16:00, Monday, November 19
(1)
(a) Let
Q
8
be the set of eight elements
{±
1
,
±
i,
±
j,
±
k
}
in the quaternion ring
H
, discussed in a
previous assignment (so
ij
=
k
=

ji
etc.). Show that
Q
8
is a group.
(b) For each of the groups
S
3
,D
4
,Q
8
do the following:
(i) Write their multiplication table;
(ii) Find the order of each element of the group;
(iii) Find all the subgroups. Which of them are cyclic?
(2)
(a) Find the order of the permutation
σ
=
±
1 2 3 4 5 6 7 8
3 1 5 6 2 7 8 4
¶
and write it as a product
of cycles.
(b) Find a permutation in
S
12
of order 60. Is there a permutation of larger order in
S
12
?
(3) Let
H
1
,H
2
be subgroups of a group
G
. Prove that
H
1
∩
H
2
is a subgroup of
G
.
(4) Let
G
be a group and let
H
1
,H
2
be subgroups of
G
. Prove that if
H
1
∪
H
2
is a subgroup then
either
H
1
⊆
H
2
or
H
2
⊆
H
1
.
(5) Let
F
be a ﬁeld. Prove that
SL
2
(
F
) :=
‰±
a b
c d
¶
:
a,b,c,d
∈
F
,ad

bc
= 1
²
is a group. If
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '07
 Goren
 Math, Algebra

Click to edit the document details