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Problem 3. Suppose that a group G acts on its power set P(G...

Problem 3. Suppose that a group G acts on its power set P(G) by conjugation. a) If H < G, prove that the normalizer of N(H) is the largest subgroup K of G such that H < K. In particular, if G is finite show that | H | divides | N(H)

Sec 45 #26. Prove that if p is an irreducible in a UFD, then...

Sec 45 #26. Prove that if p is an irreducible in a UFD, then p is a prime. Hint. Let D be a UFD, and let p be an irreducible in D. Let a, b € D such that plab .We need to show that pla or plb. If ab=0, then either a=0 and b=0 (why). We are done in this case (why). Suppose ab # 0. Then neithe...

3. In the group 224, let H = (4) and N = (6). a. List the e...

3. In the group 224, let H = (4) and N = (6). a. List the elements in HN (which we might write H + N for these additive groups) and in H n N. b. List the cosets in HN/N. showing the elements in each coset. c. List the cosets in H/(H n N), showing the elements in each coset. d. Give the correspon...

1. Use DeMoivre's Theorem to help you find a nonzero polynom...

1. Use DeMoivre's Theorem to help you find a nonzero polynomial that cos (60) = f(cos(()). 3. (a) Compute IQ(i, 3) : Q]. (b) Show that i E Q(i + 2V3). This will require some computations. (c) Show that V3 E Q(i + 2v3. (d) Show that Q(i, (3) = Q(i + 2v3) (e) Compute [Q(i+2v3::Q]. (f) Without f...

1. Let p be a prime and let G be a group with order |G| = p7...

1. Let p be a prime and let G be a group with order |G| = p7. Prove that the center of G is non-trivial, i.e. prove that there is an element 2 E G with 2 # e and such that 92 = zg for all g € G. 2. Let R be a ring and I C R an ideal. Suppose that a= b mod I and d mod I. (i) Show that a + c ...

[Q2][10 points] You are designing a rectangular poster to co...

[Q2][10 points] You are designing a rectangular poster to contain 50 in2 of printing with a 4-in. margin at the top and bottom and a 2-in. margin at each side. Set up explicitly constraint and the objective function. What overall dimensions will minimize the amount of paper used?

4. Let G be a finite group and 4 : G F [3 resp element Z...

4. Let G be a finite group and 4 : G F [3 resp element Z E FG to be 1 be a one dimensional representation. Define points Z (a) Prove that Z is an idempotent. G gEG [46-119. 2 141°3244 - X= - - (b) If R = FG, then show dimp Rx = 1. [3 poir 5. Suppose the character table of a finite gr...

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