NYU Algebra Worksheet

total of 8 questions, please see the attachment

1. Let X be an octahedron, with faces colored black and white such that no twofaces which share an edge have the same color. Let G be the group of rotationalsymmetries 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 groupG.Let G be a group which acts on a set X, and let x ∈ X. Prove that Gx = Gy forall 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, x2y}. Let D3 act on itself byleft-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(x2)ϕ(y) in S6.15. (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. Homework Four
Due on Tuesday, November 30th at 6:00pm.
For references, consult §XXI-XXVIII of the lecture notes or §6.7–7.6 of Artin.
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.
2. Let X be a right pyramid on a hexagonal base. Let Y be the solid formed by
gluing 2 copies of X together along their bases. Let G be the rotational symmetry
group of Y .
(i) Sketch Y .
(ii) Use the orbit-stabilizer theorem to find the order of G.
(iii) Use the classification of finite subgroups of SO3 (R) to determine the group
G.
3. Let G be a group which acts on a set X, and let x 2 X. Prove that Gx = Gy for
all y 2 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 ‘ : D3 ! S6 .
(ii) List the elements of S6 which form the subgroup ‘(D3 ).
(iii) Check that ‘(y)'(x) = ‘(x2 )'(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 .
6. Find the class equation for the symmetric group S5 . Give an explanation of how
you calculated each term.
7.
(i) Let G be a non-abelian group of order p3 , where p is a prime. Prove that
the center of G has order p.
(ii) Let p be a prime and let
82
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