total of 8 questions, please see the attachment
1. Let X be an octahedron, with faces colored black and white such that no two
faces which share an edge have the same color. Let G be the group of rotational
symmetries of X.
(i) Sketch X.
(ii) Let v be a vertex of X. Find |Gv| and |G(v)|.
(iii) Let e be an edge of X. Find |Ge| and |G(e)|.
(iv) Let f be a face of X. Find |Gf | and |G(f)|.
(v) Does G act transitively on vertices? On edges? On faces?
(vi) Use the orbit-stabilizer theorem to deduce the order of G.
(vii) Use the classification of finite subgroups of SO3(R) to determine the group
G.
Let G be a group which acts on a set X, and let x ∈ X. Prove that Gx = Gy for
all y ∈ G(x) if and only if Gx is a normal subgroup of G.
4. Consider the dihedral group D3 = {1, x, x2
, y, xy, x2
y}. Let D3 act on itself by
left-multiplication.
(i) Determine the associated permutation representation ϕ: D3 → S6.
(ii) List the elements of S6 which form the subgroup ϕ(D3).
(iii) Check that ϕ(y)ϕ(x) = ϕ(x
2
)ϕ(y) in S6.
1
5. (i) Partition the alternating group A4 into its conjugacy classes.
(ii) Write down the class equation for A4.
(iii) Prove that A4 is not simple.
(iv) Find the centralizer of (124) in A4.
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