Modern Algebra I
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Date Created: 10/03/15
Homework for Math 816 Fall 200809 1i DUE WEDNESDAY SEPT 10 Exercise 1 1 a Prove Theorem 121 of Chapter 1 of the notes b Compute gcd57333465 and nd integers xy E Z such that 5733 x 3465 y gcd5733 3465 c Find polynomials PX and With rational coef cients such that 2X4 7 3X2 2X PX X2 7 2 QX 1 hint Euclidls algorithm works for polynomials too Is it possible to nd such PX and With integer coef cients Exercise 12 For any integer x E Z let x be the residue class ofx in Z642Zi ls E E Z642Z X If so compute the multiplicative inverse of Hi Is 7 E Z642Z X If so compute the multiplica tive inverse of 7 Exercise 13 Let X and Y be sets and let f z X A Y be any function a Show that the relation x1 N x ltgt is an equivalence relation on Xi The equivalence classes of this relation are called the bers of b Let X denote the set of fibers of f ie the set Whose members are the equivalence classes of N and let c z X A X be the function Which takes x E X to the equivalence class of x Show that there is a unique function z X A Y such that the diagram X4gtY cl X commutes in the sense that f o c and that this function is injectivei Exercise 14 Prove Proposition 129 in Chapter 2 of the notes Exercise 15 a Suppose that dn are positive integers With d l n Show that x gt gt x is a wellde ned function ZnZ A ZdZ and that this function is a homomorphismi Explain What goes Wrong if d l n b Chinese remainder theorem Let a and b be relatively prime integersi Show that x gt gt if de nes an isomorphism ZabZ A ZaZ gtlt ZbZ c Under the isomorphism ZZlOZ A Z14Z gtlt ZlSZ which E E ZZlOZ maps to the pair 3 e Z14Z gtlt ZlSZ Exercise 16 Show that there are no integer solutions to 14X3 5Y2 ll 2i DUE WEDNESDAY7 SEPT 17 Exercise 21 Let G be a group An automorphism of G is an isomorphism f G A G The set of all automorphisms of G is denoted AutGi Prove that AutG is itself a group under composition of functions Exercise 22 Prove Proposition 134 of Chapter 2 of the notes Exercise 23 Prove Lemma 143 of Chapter 2 of the notes Exercise 24 For any a 6 Sn let co denote the number of disjoint cycles of 0 including the singleton cyclesi For example ifn 8 and a 2 5l 8 3 4 7 6 then co 4 Given a 6 Sn de ne the sign or signature of a denoted sgna7 by sgn0 47174 a Prove that every element of Sn can be written as a product of transpositions a trans position is a permutation a b which interchanges two letters and leaves the rest xed b Show that for any a 6 Sn and any transposition t 6 Sn cat ca i 1 c Now suppose that a 6 Sn and factor a t1 tT as a product of transpositionsi Prove that Sgn0 1T d Deduce that sgn Sn H i1 is a homomorphismi e Show that if a a1 a l l iak is a cycle of length k then sgno ilk1i Remark the kernel of sgn is called the alternating group An a 6 Sn l sgno 1 Exercise 25 Given two abelian groups A and B with groups laws written additively7 let HomA7 B denote the set of all homomorphisms f z A A B a Show that if 45 1 E HomA7 B then the function 45 1 de ned by lt15 1001 01 01 is again in HomA7 Bi Now show that HomAB is a group under l b Show that if A is cyclic of order n then HomACX is also cyclic of order no Exercise 26 Let n E Z be a nonzero integeri a Show that for any a E ZnZ the function multa ZnZ A ZnZ de ned by multax ax is a homomorphism7 and is an isomorphism if and only if a E ZnZ X b Prove that the function ZnZ X A AutZnZ given by a gt gt multa is a group isomorphismi 3 DUE WEDNESDAY7 SEPT 24 Exercise 31 Let X C D3 be the set containing the four reflections7 and let D3 act on X by conjugation a Compute the orbit and stabilizer of every element of X Is this action transitive b As le 4 this action determines a homomorphism D3 A 54 Compute the kernel and image of this homomorphism Exercise 32 Let Q3 denote the quaternion group of order 8 a Let Q3 act on itself by conjugation Compute the orbit and stabilizer of every element7 and the kernel of the associated homomorphism Q3 A 53 b The left regular action of Q3 on itself determines an injective homomorphism Q3 A 53 Determine the image Exercise 33 Let G be a nite group of order M and suppose that H C G is a cyclic subgroup of order n If we let H act on G by left translation7 determine the number and sizes of the orbits Exercise 34 Suppose a group G acts transitively on a set X Prove that for any two x7 y E X there is a g E G such that Staby g Stab g 1 Exercise 35 Fix a group G and for any 9 E G de ne a function conjg G A G called conjugation by g by conjgx gxg l a Prove that conjg is an automorphism of G b Prove that g gt gt conjg de nes a homomorphism G A AutG Remark this homomor phism is called the adjoint map Exercise 36 a Prove that sgn is the only nontrivial homomorphism Sn H il Hint rst show that if thtg 6 Sn are transpositions then there is a a 6 Sn such that t atla b Let Sn act on the set of n variables X1Xn by a D Xi X00 and extend this to an action of Sn on the set of all polynomials in the variables X17 Xn De ne a polynomial A H1ltiltjltnXi 7 X1 Show that the set A 7A is stable under D and that for all a 6 Sn UDAsgnaA Exercise 37 The group SL2Z A E M2Z l detA 1 acts on the set Z2 of 2 X 1 matrices For every positive integer D de ne a subset of Z2 XD gcdxy D Show that XD is equal to the orbit of under the action of SL2Z7 and deduce that two 2 v 1 lie in the same orbit if and only if gcdx7 y gcdx y Hint rst try D l vectors 4 DUE WEDNESDAY OCT 1 Exercise 41 Use the corollary of Lagrange7s theorem to deduce the little Fermat theorem if p is prime then up E a mod p for every a E Z Clearly indicate in your proof where you use the hypothesis that p is prime Exercise 42 Find all conjugacy classes of elements of order 4 in 512 you do not have to list all elements in each conjugacy class just give one representative of each c ass Exercise 43 Prove that if G is a nite group and A B C G are subgroups of relatively prime order then A N B e Exercise 44 Let 54 act by conjugation on the set X ab c where a1234 b1324 c1423 Prove that this action determines a surjective homomorphism S4 A 5X with kernel isomorphic to ZZZ gtlt ZZZ Exercise 45 Suppose n 2 5 Prove that any two 3cycles in An are conjugate Hint it suf ces to show that a l 2 3 is conjugate to 739 x y There are two cases either a and 739 are disjoint or they are not Exercise 46 a Prove that A4 and D12 are not isomorphic b Construct an isomorphism D12 2 5393 X i1 Hint let D12 act on the diagonals of the hexagon to get a map D12 A 53 then let D12 act on DlgH were H e p2 p4 739 7722 7724 is the subgroup generated by p2 and 739 Exercise 47 Let a 6 Sn be an n cycle a How big is the conjugacy class of a b Let H 739 6 Sn l 707 1 a be the set of all elements in Sn which commute with a First use FTOGA to determine lHl and then show H lt0 c Now assume that n is odd so that a E An Use part b and the FTOGA to compute the size of the conjugacy class of a in An d True or false if n is odd then all n cycles in An are conjugate meaning conjugate in An we already know they are conjugate in Sn Exercise 48 Suppose you have a hexagon and n crayons and want to color the edges of the hexagon If we regard two colorings as equivalent when they differ by a rotation how many inequivalent colorings are there Hint let X be the set of all possible colorings of the edges and let Z6Z act on X by rotation You must count the number of orbits of this action To do this rst list all subgroups of Z6Z and then determine which colorings have which subgroups as their stabilizer 5i DUE WEDNESDAY OCTi 8 Exercise 51 List all elements of order 10 in the quotient group Exercise 52 Let Q3 denote the quaternion group of order 8 a Write out the multiplication table for the quotient group Q3lt7lgti Is this quotient group isomorphic to some group you have already seen b Is there a subgroup of Q3 Which is isomorphic to Q3lt7lgt Exercise 53 Suppose that H C G is a subgroup of index 2 Show that H lt1 Gr Exercise 54 Give an example of a group G and subgroups A C B C G such that A lt1 B and lt1 C but A is not normal in Exercise 55 Suppose N lt1 G and G N lt 00 Show that if the order of x E G is relatively prime to G N then x 6 Ni Exercise 56 Let f z A A B be a surjective group homomorphism and recall that if X C A and Y C B are subgroups then so are and f 1Yi a Show that if Y C B is a subgroup then kerf C f 1B b Show that if X C A is a subgroup containing the kernel f f then f 1fX Xi c Show that if Y C B is a subgroup then ff 1Y Y d Show that if X lt1 A then lt1 B e Show that if Y lt1 B then f 1Y lt1 Al f Construct a bijection subgroups of A containing kerf A subgroups of B and that under this bijection normal subgroups correspond to normal subgroupsi Finally prove Proposition 324 of the notes Exercise 57 Let M C CX be the set of all roots of unity M ECX lHnEZsltiqnli a Show that M is a subgroup b Find a nontrivial subgroup N lt1 M With the property that MN 2 Mi Exercise 58 Suppose n 2 3 a Show that every element of An can be Written as a product of 3cyclesi b Suppose N lt1 An and that N contains at least one 3cyclei Show that N Ani 6i DUE WEDNESDAY7 OCTi 29 Exercise 61 Suppose G is a group and MN lt1 Gr a Show M N N lt1 G and that there is an injective homomorphism GM N A GM gtlt GNl b Show that if G MN then the map you constructed in a is an isomorphismi Exercise 62 Let G be a group of order 20 How many elements of order 5 does C have Exercise 63 Let G be a group of order 10 With p 4 distinct primesi Show that G is not simple Hint treat separately the cases p lt q and q lt p Exercise 64 Let G be a group of order pqr Where p lt q lt r are primesi Show that G has a normal Sylow subgroupi Exercise 65 Let P be a p Sylow subgroup of a group G and let N lt1 G be a normal subgroup of G a Show that PNN is a p Sylow subgroup of GNi b Show that P N N is a p Sylow subgroup of Ni Exercise 66 Let G be a simple group With lGl lt 60 Show that G is cyclic of prime orderi 7i DUE WEDNESDAY7 Nov 5 Exercise 71 Suppose R is a ring With 1 in which 03 13 Prove that R 03 Exercise 72 Let f R 7gt S be a homomorphism of rings a Suppose that f is an isomorphismi Show that f 1 is again a homomorphismi b Suppose that R and S are rings With 17 and that f1R 15 Show that if r E Rgtlt is a unit then E SX and fr 1 fr 1i Show by example that the conclusion need not be true Without the hypothesis f1R 15 c Show that the image off is a subring of 5 Exercise 73 Let H be the ring of quaternions and let x gt gt xL be the main involution of H de ned by x0 x1i xgj ngL x0 7 x1i7 xgj 7 xgki a Show that x E R ltgt xL x b Show that x yL xL y and xyL ytxh c Show that if x x0 x1i xgj xgk then xxL xgx x x i d The real number Nx xxL is called the norm of xi Show that Nxy e Prove that H is a division ring Exercise 74 Schoolboy binominal theorem Let p be a prime and suppose R is a commutative ring With 1 in Which p 0 Show that awry zpyp for all x7 y E R7 and conclude that x gt7 x17 de nes a ring homomorphism R 7gt R1 Exercise 75 Suppose R is a commutative ring With 1 and pX E RHXH is a power series pX a0a1Xi Show that pX is a unit in RHXH if and only if no is a unit in R1 As a corollary7 deduce Proposition 128 of the notes Exercise 76 Prove Proposition 112113 in the notes 8i DUE FRIDAY Nov 14 Exercise 81 Prove Lemma 133 in the notes Exercise 82 Prove Theorem 23 in the notes through the following sequence of steps a Consider the bijection J gt gt JI between additive subgroups of R containing I and additive subgroups of JIi Show that under this bijection J is a left ideal of R if and only if JI is a left ideal of RIi b Show that J is a maximal left ideal if and only if JI is a maximal left ideal c Assuming that R is a commutative ring With one7 show that J is a prime ideal if and only if JI is a prime ideali Exercise 83 Let R be a commutative ringi An element x E R is called nilpotent if there is a k E Z such that xk 0 A commutative ring With no nonzero nilpotents is called reduced a Prove that the set of all nilpotent elements in R7 denoted m is an ideal The ideal m is called the nilradical of Rf b Show that R is reduced c Suppose that S is a reduced commutative ring and that f R A S is a homomorphismi Show that f factors through the quotient map R A R i d Deduce that if S is any reduced quotient of R then S is isomorphic to a quotient of R i In other words7 R is the maximal reduced quotient of Rf Exercise 84 Let R be a commutative ring and suppose that P1 3 P2 3 P3 3 is a descending chain of prime idealsi Prove that the intersection 116211316 is again a prime ideali Exercise 85 Suppose that R is a commutative ring With 1 An element e E R is an idempotent if e2 1 R a Show that if e E R is an idempotent then so is f 13 7 e7 and that ef 0 b Show that if e E R is an idempotent With e f 03113 then there are nontrivial rings With 17 A and B7 such that R2 A X Bl Exercise 86 Prove Lemma 313 in the notes 9i DUE MONDAY D130 8 In this problem set D is a UFD and L FracD is its eld of fractions Let 73 C D be a complete set of irreducible representatives 0 Di Exercise 91 Suppose that D is a UFD with fraction eld L FracDi For every irreducible 7r 6 D and nonzero z E D de ne ord7rz maxk E Z l wk divides Now given any 2 E Lgtlt write 2 zy with z y E D and de ne ord7r ord7r 7 ord7r By convention ord7r0 00 a Show that ord7r Lgtlt A Z is wellde ned That is7 prove that if zy z y then ord7rzy ord7rz yi Hint look up the de nition of FracD in the notes to deter mine when two fractions are equali b Show that ord7r Lgtlt A Z is a group homomorphismi c Show that for any 2 E Lgtlt there is a unit u E Dgtlt such that 2 u H WOMAZ W673 Exercise 92 Suppose we have a nonzero polynomial E Lz of the form 101 a0a1quot39anl n a Writing each coef cient ak pkqk with pk qk E D relatively prime7 show that for every irreducible 7r 6 D minord7ra0i i i ord7ran minord7rpo7 i i i ord7rpn 7 maxord7rqo7 i i i ord7rqn hint either 7r divides some qk or it doesnlti b Deduce from a that contf H Wminordraoordran W673 recall that contf is only de ned up to multiplication by a unit in D7 so if you7re off by some u E Dgtlt then it s ok c Use the formula from b to prove Lemma 364 in the notes
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