Homework Assignments

Lecture 1:

Homework assignments for this and next week are due on Tuesday Feb 25 (not 18), due to the holiday.

Ex 10: 2,4,6,7,15,19,30-34,47
Bonus problem: Prove that the group of symmetries (resp. isometries) of the regular tetrahedron is isomorphic to A_4 (resp. S_4).

Ex 13: 1-15, 18, 28, 29, 32, 33, 38
Bonus problems:
- 1. Prove that the group of rotations of the cube is isomorphic to S_4 and that the group of isometries of the cube maps onto S_4 with kernel {1, -1}
- 2. Do the same for the dodecahedron (with S_4 replaced by A_5)

Ex 8.1: two conjugacy classes either coincide or don't intersect.
Ex 8.2: show that $K_4$ is an abelian subgroup of $S_4$ (and $A_4$).
Ex 8.3: All normal subgroups of $S_5$ are: ${e} \subset A_5 \subset S_5$
- Bonus problem: How a conjugacy class of an even element of $S_n$ splits into conjugacy classes of $A_n$.
Ex 8.4: The group $D_n$ is isomorphic to the following group of transformations of the complex plane: rotations $rho_k(z)=\epsilon^k$, where $\epsilon=e^{2\pi i/n}$, where $k=0,1,...,n-1$, and reflections $r_k(z)=\epsilon^k$ $\bar{z}$, $k=0,...,k-1$. Using this, find, all conjugacy classes of the group $D_n$ and its normal subgroups.

Ex 9.1: Let G be a group and consider the set of all non-empty subsets of G with the following binary operation: $X*Y={xy|x\in X, y\in Y}$. Show that associativity holds, e={e} is an identity element. Does the axiom of the inverse holds?
Ex 9.2: $C_mn/C_n$ is isomorphic to $C_m$.
Bonus 9.1: Any subgroup of a solvable group is a solvable group
Bonus 9.2: The braid group is generated by n-1 simple braids
From the Book:
- Ex 15: 16,19,35-37
- Ex 14: 3,7,9,10,17-21,30,34,40,41

Ex 10.1: (Q,+) is not a finitely generated group
Ex 10.2: If gcd(m,n)=1, then C_m^[n]=C_m
Ex 10.3: C_{p^n}/(C_{p^n})^[p^k] is isomorphic to C_{p^k} if n>k, and to C_{p^n} if n=k or n
From the Book:
- Ex 11: 8,10,14,15,18,20,24,29a,36,44,50
- Ex 15: 3,8,9,12

Ex 11.1 Check that the conjugation is an action
Ex 11.2 Find isotropy subgroups of and orbits of vertices under the rotation groups of the regular tetrahedron and of the cube.
Ex 11.3 Check that the map phi:G(x)-> G/G_x, phi(gx)=gG_x, is bijective
Ex 16: 1-3,8,9,11-14,18
Ex 17: 1-8

14.1 Prove that Z/nZ is a commutative associative unital ring.
Ex 18: 7-9, 11-13, 18-20, 28
Bonus problems (from Lecture 12)
- 12.1 In how many ways can the faces of a regular tetrahedron be couloured by 3 colours?
- 12.2. Prove Sylow's 1st Theorem if |G|=pg, where p and g are prime.
- 12.3. Prove that any group of order 2q, where q is an odd prime, is isomorphic either to C_2q or to D_q

From the book:
- Ex 18: 22,23,25,28
- Ex 20: 1,2,4,5,6,12,19
- Ex 16.1. IN Z_12[x] the polynomial x^5-5x+6 has 4 roots: 2,3,6,11
Bonus
- 16.1. let P(x) be a non-zero polynomial of degree n in D[x], where D is a domain. Show that P(x) has at most n roots.
- 16.2. If G is an abelian non-cyclic group of order n, then there exists a positive integer m<n, such that g^m=e for all g in G.
- 16.3. Show that Z^x_{2^n} is isomorphic to C_2xC_2^{n-2}, where C_2 is generated by 2^n -1 and C_2^{n-2} is generated by 5. Show that Z^x_{p^k] is isomorphic to the direct product cyclic subgroup, generated by p+1 (which has order p^{k-1} and the cyclic subgroup, generated by a, where a is a generator of Z^x_p (which has order p-1. In particular this group is cyclic.

From the book:
- Ex 22: 5,12,13,17,23
- Ex 23: 1,7,9,12,14,16,21
Bonus 18.1: Prove that the set of quaternions {\pm 1, \pm i, \pm j, \pm k} w.r.to multiplication form a non-abelian group of order 8. Show that this group and D_4 are the only non-abelian groups of order 8.

There'll be no class on April 29, insted there'll be 2 classes on May 1

Exercise 19.1. Show that in the ring F[x,y] the ideal I of polynomials with zero constant term is not a principal ideal.
Exercise 19.2. Show that a unital commutative ring having only trivial ideals is a field.
From the book:
- Ex 27: 2,4,6,10,11,15-18
Bonus 19.1: Prove that over any finite field F_q and any positive integer n there exists an irreducible polynomial over F_p of degree n.

Reminder: There'll be no class on April 29, insted there'll be 2 classes on May 1

Exercise 20.1. Show that if (a) is a maximal ideal in a PID, then a is an irreducible element.
Ex 45: 1,3,5,9,10,18-21, 27,30-32
Bonus 20.1: If R is a factorizable domain, then R[x] is a factorizable domain as well.