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3.2. SEMIDIRECT PRODUCTS
155
that
'
W
x
7!
.'
1
.x/;:::;'
r
.x//
is an injective homomorphism
into
A
1
± ²²² ±
A
r
.
(b)
Explore conditions for
'
to be surjective.
3.1.19.
Let
N
1
;N
2
;:::;N
r
be normal subgroups of a group
G
. Show that
N
i
\
.N
1
:::N
i
±
1
N
i
C
1
:::N
r
/
D f
e
g
for all
i
if, and only if, whenever
x
i
2
N
i
for
1
³
i
³
r
and
x
1
x
2
²²²
x
r
D
e
, then
x
1
D
x
2
D ²²² D
x
r
D
e
.
3.2. Semidirect Products
We now consider a slightly more complicated way in which two groups
can be ﬁt together to form a larger group.
Example 3.2.1.
Consider the dihedral group
D
n
of order
2n
, the rotation
group of the regular
n
gon.
D
n
has a normal subgroup
N
of index 2 con
sisting of rotations about the axis through the centroid of the faces of the
n
gon.
N
is cyclic of order
n
, generated by the rotation through an angle
of
2±=n
.
D
n
also has a subgroup
A
of order 2, generated by a rotation
j
through an angle
±
about an axis through the centers of opposite edges (if
n
is even), or through a vertex and the opposite edge (if
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 Fall '08
 EVERAGE
 Algebra

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