Fall 2019, Math 463 - Abstract Algebra - Mathematics & Statistics

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Posted: April 19, 2024 - 1:32 PM

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Fall 2019, Math 463 – Abstract Algebra

The syllabus.

Homework #1, due Fri., Sept. 20:
Judson, Section 1.3: #6, 10, 11, 17ad, 21, 22ac, 24bde, 28

Homework #2, due Fri., Sept. 27:
Problem 1: By analyzing the multiplication table (Cayley table), show that (Z/5Z,+) is the only group of order 5, up to renaming the elements of the set.
Goodman: #1.1.1, 1.1.2, 1.3.2, 1.3.3
Judson, Section 3.4: #2, 3, 10, 16, 25 (induction!), 31, 32, 33

Homework #3, due Weds., Oct. 9:
Judson, Section 2.3: 7, 15e, 17ae, 22 (I sketched this in class; I’m asking you to fill in the details), 27, 28, 30 (hint: a product of numbers =1 mod 4 is also =1 mod 4), 31
Goodman: 1.6.3, 1.6.7, 1.7.13 (“invertible” here means “has a multiplicative inverse”; see page 41), 1.7.16
Hint for Judson 15e and Goodman 1.7.16: Some of your classmates know how to do this, ask them; you may not use trial-and-error.

Homework #4, due Weds., Oct. 23:
Problem 1: List all of the cyclic subgroups of S₄ (please don’t duplicate entries/subgroups), and give the order of every element of S₄. Please don’t start this problem (or the next) until we’ve talked about “cycle notation” for permutation groups. (And please express your answers using cycle notation.)
Problem 2: List (any) two proper non-cyclic subgroups of S₄.
Judson, Section 3.4: 46, 52, 53, 54
Judson, Section 4.4: 1de, 4f, 10, 11, 23c, 37, 45
Goodman: 2.2.15, 2.2.16, 2.2.19, 2.2.28 (Φ(n) is (Z/nZ)^x, see Lemma 1.10.3)

Homework #5, due Fri., Nov. 1:
Judson, Section 5.3: 1, 2abdjo, 5 (only list the non-cyclic subgroups), 32
Judson, Section 9.3: 9, 37, 38, 41, 42
Judson, Section 11.3: 2, 5, 11
Goodman: 1.5.8, 2.4.14

Homework #6, due Fri., Nov. 15:
Problem 1: Use the construction in Cayley’s Theorem to find a subgroup of S₁₀ isomorphic to D_2*5 (the dihedral group of order 10). Write the subgroup in cycle notation.
Judson, Section 9.3: 2, 11, 16, 17 (Judson’s D₄ is my D_2*4, the dihedral group of order 8), 18 (your proof should be at most two sentences), 22 (also list all possible groups of this type, up to isomorphism), 26, 28, 48, 50

Homework #7, due Fri., Dec. 6:
Judson 6.4: 5ab, 6, 7, 8, 17
Judson 10.3: 1abd, 2, 14
Judson 11.3: 13, 14, 15, 16
Goodman 2.5: 2.5.2, 2.5.10
Goodman 2.7: 2.7.4, 2.7.6cd (we’ve already done parts a and b, Int(G) is usually called Inn(G)), 2.7.8, 2.7.11
Goodman 3.2: 3.2.2, 3.2.6

Homework #8, due Fri. Dec. 13:
Judson 14.4: 2 (note Judson’s X_g is my X^{g}), 3, 4, 5
Goodman 5: 5.1.5, 5.1.6 (already proved G_x is a subgroup)

All redos are due Fri. Dec. 13, as well.