Exercise 1
Due March 24 1999
1. Prove that the group of symmetries of the tetrahedron is isomorphic
to S4 in two ways:
-
Find an explicit isomorphism beteween the group of symmetries of
the tetrahedron and the group Sym(corners) of permutations of the corners
of the tetrahedron. Which subgroup of Sym(corners) corresponds to the rigid
motions of the tetrahedron?
-
On each face of the cube, draw one diagonal red and the other one blue
in such a way that the 6 red diagonals form the edges of a regular tetrahedron
and the 6 blue diagonals from the edges of another regular tetrahedron.
Show that the group of symmetries of the cube acts on the set {red tetrahedron,
blue tetrahedron} and the kernel of this action acts as the group of symmetries
of the red tetrahedron. Deduce that the group of symmetries of the tetrahedron
is isomorphic to the group of rigid motions of the cube.
2. Let (X,G) - > X be a group action. For each x in X, Gx
is the subset of G which fixes x. Show that Gx is a subgroup
of G.
Let K be the kernel of the action. Prove the following:
-
K= intersection{x in X} Gx
-
If the action is transitive on X, then for each x in X,
K= intersection{a in G} a-1
Gx a
3. The circle group T = {z in C : |z|=1 }, where
C is the field of complex numbers, and the operation is multiplication
of complex numbers.
-
Show that T is isomorphic to the group of orthogonal matrices of
determinant 1 over R, the real numbers, and to the group R/Z,
the factor group of the additive group of real numbers by the additive
group of integers.
For solutions, see this page. To return to
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click here .
Last update March 24, 2000
Author: Phill Schultz, schultz@maths.uwa.edu.au