PROBLEM SETS
Assignment 1
Ex 2: 1-4, 7-10, 12
Ex 4: 1-6, 12, 14, 20
Ex 5: 1-3, 13, 20
Ex 6: 4, 17, 19, 21, 22, 28, 32, 33, 34
Bonus problems:
1) If associativity of 3 factors holds, then prove that a*b*c*d is independent of the way one puts parentheses.
2) Prove that bij(S) is commutative if and only if |S| = 1 or 2.
Assignment 2
Ex 6: 38, 41, 46, 48, 55
Ex 7: 1, 3, 5, 6
Ex 8: 2, 8, 10, 17, 23, 26, 30
Ex 9: 1, 10, 14
Bonus problems:
1) Find all continuous isomorphisms: (R,+) --> (R_>0,x)
2) Classify upto an isomorphism, all groups order 1, 2, 3 and 4 the cyclic group usually denoted by C_n (or order n), show that for n>=1, there is a group of order n. Show that for n=1, 2, 3 there are no other groups of order but for n=4 there is just one non-ismorphic to C_4.
3) Find an algorithm for determining the k of the Chinese Remainder Theorem, other than just checking all numbers between zero and the product of all m_i where i goes from 1 to n.
4) Show if two cycles intersect then they do not commute.
5) Find the subgroup of S_4 which is the group of symmetries of the regular tetrahedron.
Assignment 3
Ex 8: 44, 45, 47
Ex 9: 7, 11, 13abc, 15, 23
Bonus problems:
1)Prove the following:
a)The group of orientation preserving symmetries of a regular tetrahedron is isomorphic to A_4 and of a cube to S_4.
b)The group of all symmetries of a tetrahedron is isomorphic to S_4.
c)A harder problem: the group of orientation preserving symmetries of a regular icosahedron is isomorphic to A_5.
2)Prove that all finite groups G of symmetries of a plane are isomorphic either to C_n for n>=1 or to D_n. (Hint: Prove that G has fixed point.)
3)Prove that the Braid group B_n is a group.
Assignment 4
Ex 10: 2, 4, 6, 7, 15, 19, 30, 31, 32, 33, 34, 47
Ex 13: 1-15, 18, 28, 29, 32, 33, 38
Bonus problems:
1)
2)If n exists in Z and the function f_n(a)=a^n, where a exists in a group G, then prove that f_n:G-->G is a homomorphism if and only if G is abelian.
Assignment 5
Ex 14: 5, 7, 9, 23, 26
Ex 15: 16, 35, 36, 37
Ex 11: 8, 10, 14, 20, 24, 29a, 36, 44, 50
Bonus problems:
1) In S_5, the only non-trivial normal subgroup is A_5.
2) Describe for which cycle decomposition types in A_n, the conjugacy clas consists of an entire conjugacy class of S_n
3) A_4 (where order is 12) contains no subgroup of order 6.
4)
Assignment 6
Ex 16: 1, 2, 3, 8, 9, 11, 12, 13, 14, 18
Ex 17: 1, 2, 3, 4, 5, 6, 7, 8
Bonus problems:
1) In the group of isometries of R the set of elements of finite order are reflection. Show that the product of two at distinct points is a translation if it has infinite order.
2) (C_(pk))^[p] ~- (C_(pk-1))
3) There si one and only one group of order 8 outside the list denoted by Q.
4) How many real dice are there? (The sum of opposite faces is 7)
5) Finish the computation being done in class. How many different dice can one have?
Assignment 7
Ex 18: 7,8,9,11,12,13,18,19,20,28
Ex 19: 1,2,14,17,18,19
Ex 20: 3,4,10,20,24
Bonus problems:
1) Prove that 1 is unique if it exists.
2) Prove that if G is a non-cyclic abelian group of order n, then there exists 1< k< n such that a^k=e for all a in G.
3) Prove that a finite domain is always a field.
Assignment 8
Ex 22: 5, 12, 13, 17, 23
Ex 23: 1, 7, 9, 12, 14, 16, 21
Ex 24: 4, 6, 8, 9, 10, 11
Ex 26: 1, 3, 10, 13, 14, 17, 18, 38
Bonus problems:
1) (R_1XR_2)^x = R_1^x X R_2^x
2) If p is a prime > 2, let Z_p^x = < a>. Prove that Z_(p^k)^x is isomorphic to
3) Z_(2^k)^x = {+-1} X <5> if k>= 2 which is not cyclic. [{+-1} is isomorphic to C_2 and <5> is isomorphic to C_(2^(k-2))]
4) Decompose x^4 + 1 =0 as a product of quadratic polynomials.
5) The only commutative division algebra over R are R and C
6) (a/-b . b/a)(a_1/-b_1 . b_1/a_1) again has this form det(a/-b . b/a)=|a^2|+|b^2|does not equal 0 unless (a/-b . b/a)=0. So any non-zero element in H is invertible, so H is a division algebra.
7) i^2=-1, j^2=-1, k^2=-1, ij=k, jk=i, ki=j, ij=-ji, ik=-ki, jk=-kj. Check all these relations and show that they follow from the following set of Hamiltonian relations: i^2=j^2=k^2=ijk=-1, show also: a+bi+cj+dk=a-bi-cj-dk, (qq_1)^--= q^- . q_1^-, |a+bi+cj+dk|^2=a^2+b^2+c^2+d^2
Assignment 9
Ex 27: 2,4,6,10,11,15,16,17,18
Ex 45: 1,3,5,9,10,18,19,20,21,27,30,31,32
Bonus problems:
1) Show that the image of a ring R under a homomorphism f:R-->R' is a subring of R'.
2) Check that R/I with well-defined addition and multiplication is a ring.
3) Prove the fundamental homomorphism theorem for rings.
Assignment 10
Ex 46: 1,2,13,15,17,18,20
Ex 47: 1,2,5,8,15,16,18
Assignment 11
Ex 45: 14, 17, 29, 15
Ex 46: 9, 12, 13(not required, already done in previous HW), 21
Ex 29: 4, 5, 6, 8, 10, 11, 12, 18, 23, 25
Ex 31: 2, 3, 4, 11, 19, 23, 24, 26, 27
Bonus problems:
1) Construct infinitely many rings Z[\sqrt{-m}] where m is square free, which are not UFD.
2) Recall that C-~ {a/-b . b/a | a, b exist in R}, H-~{a/-b . b/a | a, b exist in C}, A-~{a/-b . b/a | a, b exist in H}. Why is tis not a division algebra?
3) Show that for all p, a positive prime integer, there exists a finite field with p^n elements, that is for any n, there exists an irreducible polynomial over Z_p of degree n.
4) Show that x^4-2x^2-1 is irreducible over Q.
