MA249 - Algebra II

MA249 - Algebra IIAssignment 1 January 2008Answer the questions on your own paper. Write your own name in the top left-handcorner, and your university ID number in the top right-hand corner. Use the problemsat the beginning as well as exercises in the lecture notes for a warm up. Solutions to theFOUR TEST problems must be handed in by 15.00 on MONDAY 21 JANUARY(Monday of the third week of term), or they will not be marked. There will be an awardof 5 extra marks for clarity, so do a good job.These are practice problems for you to sharpen your teeth on.P1. Find all subgroups of the group C 6 .P2. Find the orders of all elements in the group C 12 . How do you find the order of anelement in C n for general n?P3. Prove that a group G is abelian if and only if f : G → G defined by f(x) = x 2 is agroup homomorphism.P4. Let I be a left ideal of a ring R. Prove that I = R if and only if 1 ∈ I.P5. Let us consider a set X. Let 2 X be the set of all subsets of X. We define amultiplication on 2 X as the intersection and an addition as the symmetric difference:A · B = A ∩ B, A + B = (A ∪ B) \ (A ∩ B).Prove that 2 X is a commutative ring under these operations.P6. Let G = Sym(X). Let Y be a subset of a set X. Show that the subset G Y of Gdefined by G Y = {g ∈ G | g(y) ∈ Y and g −1 (y) ∈ Y for all y ∈ Y } is a subgroup of G.If X is finite, what is the order of G Y as a function of |Y | and |X|?P7. Consider a ring R. Define ̂R = R × Z with operations given by (r,n) + (s,m) =(r + s,n + m) and (r,n) · (s,m) = (rs + mr + ns,mn). Prove that ̂R is a ring.The following problems are test problems for you to submit for marking. Writeconcise but complete solutions only to the questions asked. My solution to each problemtakes half of A4 page. Your solution should take one A4 page and if it is taking more thantwo then you are writing too much.1. Show that the order of an element g ∈ Sym(X) is the least common multiple of thelengths of the cycles of g, when g is written in cyclic notation, that is, written as a productof disjoint cycles.Elements of largest possible orders in S 3 , S 4 , S 5 are (1,2,3) – order 3, (1,2,3,4) – order 4,and (1,2,3)(4,5) – order 6. Find elements of largest possible orders in S n for 6 ≤ n ≤ 12.[5 marks]2. We consider a group G in this problem with the property that g 2 = 1 for all g ∈ G.Prove that G is abelian.Show that G admits a vector space structure over the field Z 2 of 2 elements. (Hint: youneed to define addition of vectors and multiplication by a scalar. There is only one binaryoperations ltying around. Then check the axioms.)Prove that if G is finite then G contains 2 n elements for some natural n. (Hint: have youheard of a basis?)[5 marks]

3. Let A be an abelian group. Let E be the set of all homomorphisms from A to A, thatis E = {f : A → A | f(ab) = f(a)f(b) for all a,b ∈ A}. Prove that E is a ring under thefollowing operations:f + g : x ↦→ f(x)g(x), fg : x ↦→ f(g(x)).Let A = G be the abelian group in problem 2 (with g 2 = 1 for each element). Show thatE is equal to the set of all linear transformations from A to A once you consider A as avector space over Z 2 .Furthermore, assume that G is finite. In problem 2, we have proved |G| = 2 n for some n.Prove that |E| = 2 n2 .[4 marks]4. Consider the standard 3-dimensional vector space V = R 3 with operations × and • ofvector and scalar product. The quaternions are elements of a vector space H = R ⊕ V .They are a bit like complex numbers: α is a real part of a quaternion (α,v), and v is itsimaginary part. Quaternions with zero real part are called imaginary.The quaternions form a ring under the addition in the vector space and the multiplicationdefined by formula (α,v) ·(β,w) = (αβ −v •w,αw +βv +v ×w). You don’t have to proveit here and you are allowed to use this fact later.Pick an orthonormal basis I, J, K of V . Show that the 8 elements ±I, ±J, ±K, and±1 H form a group under multiplication and write explicitly the multiplication table inthis group. This group is called quaternion group and denoted Q 8 . (Hint: you can deduceassociativity from the fact H is a ring that you are allowed to use.)We define the norm on H by |(α,v)| = √ α 2 + |v| 2 . You can use the fact that the norm ismultiplicative, that is, satisfies |(α,v) · (β,w)| = |(α,v)||(β,w)|. The complex conjugationof quaternions is defined by (α,v) ∗ = (α, −v). Prove that hh ∗ = (|h| 2 ,0) for any h ∈ H.Show that all elements of norm 1 form a group under multiplication of quaternions. Notice(but don’t prove) that geometrically it is a 3-sphere S 3 .Show that x 2 = −1 if and only if x is imaginary of norm 1. Notice that such quaternionsform a 2-sphere S 2 .Consider a map π : S 3 → S 2 defined by π(q) = qIq −1 for all q ∈ S 3 . Show that the inverseimage π −1 (I) is 1-sphere (a circle) S 1 .In fact, π is surjective with all inverse images being S 1 (you don’t need to prove it). Thisis called Hopf fibration and it is a very nice way to imagine what S 3 looks like.[6 marks]