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
1. (a) Let G be a group and consider the power set P(G) = {SCG) Explicitly...
Consider the rotational symmetry group G of the cube Let X be the set of edges of the cube, and let xe X be the edge between faces A and E (see picture). G acts on X in the obvious way. Describe the stabilizer Stabg(x) and the orbit Orbg(x). By using the orbit-stabilizer theorem, deduce G.
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...
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:...
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:...
1. Let G be the group from our warmup that's located on page 146: G = {e, (132)(456)(78),(132)(465), (123)(456) (123)(456)(78),(78)}. This problem is designed to verify that the orbit-stabilizer theorem is true, and that the factors involved can vary. (a) (5 points) Compute orbç(3). (b) (5 points) Compute stabc(3). (c) (5 points) Verify that lorbo(3). Istabo(3) = G). (d) (5 points) Compute orbG(8). (e) (5 points) Compute stabg(8) (1) (5 points) Verify that forbo(8)|- |stabo(8)| = GL. (8) (5 points)...
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...
Question 4
Exercise 1. Let G be a group such that |G| is even. Show that there exists an EG,17e with x = e. Exercise 2. Let G be a group and H a subgroup of G. Define a set K by K = {z € G war- € H for all a € H}. Show that (i) K <G (ii) H <K Exercise 3. Let S be the set R\ {0,1}. Define functions from S to S by e(z)...
. (15 points) Let G be a group and A be a nonempty subset of G. Consider the set Co(A) = {9 € G gag- = a for all a € A}. (a) Compute Cs, ({€, (123), (132)}), where e is the identity permutation. (b) Show that CG(A) is a subgroup of G. (c) Let H be a subgroup of G. Show that H is a subgroup of Ca(H) if and only if H is abelian.
Q16
S CX}. If G has a group 15. The powerset of a set, X, is defined to be the collection of all subsets of X: P(X) = { S action on X, then the group action can be defined on P(X) by a. S = {a.s | SES}. (a) Show that if S = orb(r), then a.S= S for all a E G. (b) If a. S = S, show that S = U; orb(r) for some elements r;...
4. If G is a group, then it acts on itself by conjugation: If we let X = G (to make the ideas clearer), then the action is Gx X = (g, x) H+ 5-1xg E G. Equivalence classes of G under this action are usually called conjugacy classes. (a) If geG, what does it mean for x E X to be fixed by g under this action? (b) If x E X , what is the isotropy subgroup Gx...