120
2. BASIC THEORY OF GROUPS
(a)
For
A
2
GL
.n;
R
/
and
b
2
R
n
, define the transformation
T
A;
b
W
R
n
!
R
n
by
T
A;
b
.
x
/
D
A
x
C
b
. Show that the set of all such
transformations forms a group
G
.
(b)
Consider the set of matrices
A
b
0
1
;
where
A
2
GL
.n;
R
/
and
b
2
R
n
, and where the
0
denotes a 1
by
n
row of zeros. Show that this is a subgroup of GL
.n
C
1;
R
/
,
and that it is isomorphic to the group described in part (a).
(c)
Show that the map
T
A;
b
7!
A
is a homomorphism from
G
to
GL
.n;
R
/
, and that the kernel
K
of this homomorphism is iso
morphic to
R
n
, considered as an abelian group under vector ad
dition.
2.4.21.
Let
G
be an abelian group. For any integer
n > 0
show that the
map
'
W
a
7!
a
n
is a homomorphism from
G
into
G
. Characterize the
kernel of
'
. Show that if
n
is relatively prime to the order of
G
, then
'
is
an isomorphism; hence for each element
g
2
G
there is a unique
a
2
G
such that
g
D
a
n
.
2.5. Cosets and Lagrange’s Theorem
Consider the subgroup
H
D f
e; .1 2/
g
S
3
. For each of the six elements
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 Fall '08
 EVERAGE
 Algebra, Transformations, Matrices, Vector Space, TA, Abelian group

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