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Consider the rotational symmetry group G of the cube Let X be the set of edges...
Let Tetra act on the set S of pairs of opposite edges of the tetra- hedron. What is the stabilizer of one such p...
Let Tetra act on the set S of pairs of opposite edges of the tetra- hedron. What is the stabilizer of one such pair? Use Corollary 1.4 to recompute |Tetral. 1. Corollary 1.4. If a finite group G acts on a set S, then for any SES, IG| = #(Os)|G5|. In particular, # (0s) divides |G|.
Let Tetra act on the set S of pairs of opposite edges of the tetra- hedron. What is the stabilizer of one such pair?...
2. Assume the group G acts on the set S. For E S, define Then G is a subgroup of G , which is called the stabilizer of r. The set is called the orbit of r (a) Consider the map ф' G S, defined...
2. Assume the group G acts on the set S. For E S, define Then G is a subgroup of G , which is called the stabilizer of r. The set is called the orbit of r (a) Consider the map ф' G S, defined by фг (g) :-9-x. Prove that there is one map (and only one) : G/G, S such that Vz ยู่'z q (where q: G -G/G, is the quotient map). (b) Prove that is injective. (Hint:...
Let G be a group of order 6 and let X be the set (a, b,c) E G3: abc That is, X is the set of trip...
Let G be a group of order 6 and let X be the set (a, b,c) E G3: abc That is, X is the set of triples of elements of G with the product of its coordinates equals the identity element of G (a) How many elements does X have? Hint: Every triple (a, b, c) in X is completely determined by the choice of a and b. Because once you choose a and b then c must be (ab)-1...
1. (a) Let G be a group and consider the power set P(G) = {SCG) Explicitly...
1. (a) Let G be a group and consider the power set P(G) = {SCG) Explicitly verify that GXP(G) + P(G) (9,8) gSg-1 = {gsg- S ES} is a group action of G on P(G). (b) Let G = Ss, and consider the subset S = ((1 3), (25)) E P(G). Compute the orbit of S under the action of G, as well as the stabilizer of S in G.
2. Assume the group G acts on the set S. For E S, define Then G...
2. Assume the group G acts on the set S. For E S, define Then G is a subgroup of G , which is called the stabilizer of r. The set is called the orbit of r (a) Consider the map ф' G S, defined by фг (g) :-9-x. Prove that there is one map (and only one) : G/G, S such that Vz ยู่'z q (where q: G -G/G, is the quotient map). (b) Prove that is injective. (Hint:...
answer question 3 , imagine putting a small cube inside a larger cube and let G be the graph whose vertex set consists of the 8 corners of the small cube and the 8 corners of the larger cube (1...
answer question 3
, imagine putting a small cube inside a larger cube and let G be the graph whose vertex set consists of the 8 corners of the small cube and the 8 corners of the larger cube (16 vertices total) and whose edge set consists of the edges in each cube (12 per cube) and the edges joinin corresponding vertices of the two cubes (8 more), for a total of 32 edges. 3. Find a Hamilton Circuit in...
read exercise and do question 6 In problems 3 through 6, imagine putting a small cube inside a larger cube and let G be the graph whose vertex set consists of the 8 corners of the small cube an...
read exercise and do question 6
In problems 3 through 6, imagine putting a small cube inside a larger cube and let G be the graph whose vertex set consists of the 8 corners of the small cube and the 8 corners of the larger cube (16 vertices total) and whose edge set consists of the edges in each cube (12 per cube) and the edges joining corresponding vertices of the two cubes (8 more), for a total of 32...
R-(003,0340 2. Let X-1,2,3, 4, 5,6) and let G (1), (123), (132), (45), (123) (45), (132) (45)). L...
r-(003,0340 2. Let X-1,2,3, 4, 5,6) and let G (1), (123), (132), (45), (123) (45), (132) (45)). Let G act on X in the obvious way. (a) For eachx X and g E G, find Ox, G and Xg. Label these clearly. (b) Verify the orbit-stabilizer theorem and Burnside's lemma for this example and explain (i.e., demonstrate that you know what these are and mean). c) To thank your professors for doing such an amazing job all semester, you decide...
Consider the rectangle shown, and let A be the eight points listed. The symmetry group, G, of thi...
Consider the rectangle shown, and let A be the eight points listed. The symmetry group, G, of this rectangle has four elements: the identity j a flip over a horizontal axis through its centre v a flip over a vertical axis through its centre r a rotation about its centre by 180 degrees. We regard G as a subgroup of SA Page 2 (a) Find G(a) and G(b), the orbits of a and b. (b) Find Ga and Gb, the...
Exercise 2. Let he a group anith nentral element e. We denote the gronp lau on G simply by (91,92...
Exercise 2. Let he a group anith nentral element e. We denote the gronp lau on G simply by (91,92)gig2. Let X be a set. An action ofG on X is a a map that satisfies the following tuo conditions: c. Let G be a finite group. For each E X, consider the map (aje- fer all elements r X (b) 9-(92-2) for all 91,92 G and all r E X Show that is surjective and that, for all y...
Let Tetra act on the set S of pairs of opposite edges of the tetra- hedron. What is the stabilizer of one such pair? Use Corollary 1.4 to recompute |Tetral. 1. Corollary 1.4. If a finite group G acts on a set S, then for any SES, IG| = #(Os)|G5|. In particular, # (0s) divides |G|.
Let Tetra act on the set S of pairs of opposite edges of the tetra- hedron. What is the stabilizer of one such pair?...
2. Assume the group G acts on the set S. For E S, define Then G is a subgroup of G , which is called the stabilizer of r. The set is called the orbit of r (a) Consider the map ф' G S, defined by фг (g) :-9-x. Prove that there is one map (and only one) : G/G, S such that Vz ยู่'z q (where q: G -G/G, is the quotient map). (b) Prove that is injective. (Hint:...
Let G be a group of order 6 and let X be the set (a, b,c) E G3: abc That is, X is the set of triples of elements of G with the product of its coordinates equals the identity element of G (a) How many elements does X have? Hint: Every triple (a, b, c) in X is completely determined by the choice of a and b. Because once you choose a and b then c must be (ab)-1...
1. (a) Let G be a group and consider the power set P(G) = {SCG) Explicitly verify that GXP(G) + P(G) (9,8) gSg-1 = {gsg- S ES} is a group action of G on P(G). (b) Let G = Ss, and consider the subset S = ((1 3), (25)) E P(G). Compute the orbit of S under the action of G, as well as the stabilizer of S in G.
2. Assume the group G acts on the set S. For E S, define Then G is a subgroup of G , which is called the stabilizer of r. The set is called the orbit of r (a) Consider the map ф' G S, defined by фг (g) :-9-x. Prove that there is one map (and only one) : G/G, S such that Vz ยู่'z q (where q: G -G/G, is the quotient map). (b) Prove that is injective. (Hint:...
answer question 3
, imagine putting a small cube inside a larger cube and let G be the graph whose vertex set consists of the 8 corners of the small cube and the 8 corners of the larger cube (16 vertices total) and whose edge set consists of the edges in each cube (12 per cube) and the edges joinin corresponding vertices of the two cubes (8 more), for a total of 32 edges. 3. Find a Hamilton Circuit in...
read exercise and do question 6
In problems 3 through 6, imagine putting a small cube inside a larger cube and let G be the graph whose vertex set consists of the 8 corners of the small cube and the 8 corners of the larger cube (16 vertices total) and whose edge set consists of the edges in each cube (12 per cube) and the edges joining corresponding vertices of the two cubes (8 more), for a total of 32...
r-(003,0340 2. Let X-1,2,3, 4, 5,6) and let G (1), (123), (132), (45), (123) (45), (132) (45)). Let G act on X in the obvious way. (a) For eachx X and g E G, find Ox, G and Xg. Label these clearly. (b) Verify the orbit-stabilizer theorem and Burnside's lemma for this example and explain (i.e., demonstrate that you know what these are and mean). c) To thank your professors for doing such an amazing job all semester, you decide...
Consider the rectangle shown, and let A be the eight points listed. The symmetry group, G, of this rectangle has four elements: the identity j a flip over a horizontal axis through its centre v a flip over a vertical axis through its centre r a rotation about its centre by 180 degrees. We regard G as a subgroup of SA Page 2 (a) Find G(a) and G(b), the orbits of a and b. (b) Find Ga and Gb, the...
Exercise 2. Let he a group anith nentral element e. We denote the gronp lau on G simply by (91,92)gig2. Let X be a set. An action ofG on X is a a map that satisfies the following tuo conditions: c. Let G be a finite group. For each E X, consider the map (aje- fer all elements r X (b) 9-(92-2) for all 91,92 G and all r E X Show that is surjective and that, for all y...
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