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262
5. ACTIONS OF GROUPS
5.6.5.
Let
G
be the rotation group of the cube, acting on the set of faces of
the cube. Show that map
F
7!
Stab
.F/
is 2to1, from the set of faces to
the set of stabilizer subgroups of faces.
5.6.6.
Let
G
D
S
n
and
H
D
Stab
.n/
Š
S
n
±
1
. Show that
H
is its own
normalizer, so that the cosets of
H
correspond 1to1 with conjugates of
H
. Describe the conjugates of
H
explicitly.
5.6.7.
Identify the group
G
of rotations of the cube with
S
4
, via the action
on the diagonals of the cube.
G
also acts transitively on the set of set of
three 4–fold rotation axes of the cube; this gives a homomorphism of
S
4
into
S
3
.
(a)
Compute the resulting homomorphism
of
S
4
to
S
3
explicitly.
(For example, compute the image of a set of generators of
S
4
.)
Show that
is surjective. Find the kernel of
.
(b)
Show that the stabilizer of each 4–fold rotation axis is conjugate
to
D
4
±
S
4
.
(c)
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 Fall '08
 EVERAGE
 Algebra

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