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188 CHAPTER 10 MATRIX GROUPS AND SYMMETRY(b) The quaternion group, Q 8 , cannot be written as a semidirect product.(c) E(2) is the semidirect product of O(2) by H, where H consists of alltranslations in R 2 .16. Determine which of the 17 wallpaper groups preserves the symmetry of thepattern in Figure 10.5.Figure 10.11.17. Determine which of the 17 wallpaper groups preserves the symmetry of thepattern in Figure 10.11.18. Find the rotation group of a dodecahedron.19. For each of the 17 wallpaper groups, draw a wallpaper pattern having thatgroup as a symmetry group.References and Suggested Readings[1] Coxeter, H. M. and Moser, W. O. J. Generators and Relations for DiscreteGroups, 3rd ed. Springer-Verlag, New York, 1972.[2] Grove, L. C. and Benson, C. T. Finite Reflection Groups. 2nd ed. Springer-Verlag, New York, 1985.[3] Hiller, H. “Crystallography and Cohomology of Groups,” American MathematicalMonthly 93 (1986), 765–79.[4] Lockwood, E. H. and Macmillan, R. H. Geometric Symmetry. CambridgeUniversity Press, Cambridge, 1978.[5] Mackiw, G. Applications of Abstract Algebra. Wiley, New York, 1985.[6] Martin, G. Transformation Groups: An Introduction to Symmetry. Springer-Verlag, New York, 1982.[7] Milnor, J. “Hilbert’s Problem 18: On Crystallographic Groups, FundamentalDomains, and Sphere Packing,” Proceedings of Symposia in Pure Mathematics18, American Mathematical Society, 1976.[8] Phillips, F. C. An Introduction to Crystallography. 4th ed. Wiley, New York,1971.
EXERCISES 189[9] Rose, B. I. and Stafford, R. D. “An Elementary Course in MathematicalSymmetry,” American Mathematical Monthly 88 (1980), 54–64.[10] Schattschneider, D. “The Plane Symmetry Groups: Their Recognition andTheir Notation,” American Mathematical Monthly 85 (1978), 439–50.[11] Schwarzenberger, R. L. “The 17 Plane Symmetry Groups,” MathematicalGazette 58 (1974), 123–31.[12] Weyl, H. Symmetry. Princeton University Press, Princeton, NJ, 1952.
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Abstract AlgebraTheory and Applicat
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PrefaceThis text is intended for a
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PREFACEixappears at the end of each
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CONTENTSxi6 Introduction to Cryptog
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0PreliminariesA certain amount of m
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0.1 A SHORT NOTE ON PROOFS 3ax 2 +
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0.2 SETS AND EQUIVALENCE RELATIONS
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0.2 SETS AND EQUIVALENCE RELATIONS
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10 CHAPTER 0 PRELIMINARIESnot all f
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12 CHAPTER 0 PRELIMINARIESThis is a
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14 CHAPTER 0 PRELIMINARIESgiven by
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16 CHAPTER 0 PRELIMINARIESandB =the
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18 CHAPTER 0 PRELIMINARIESIf we con
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20 CHAPTER 0 PRELIMINARIES(e) If g
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1The IntegersThe integers are the b
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24 CHAPTER 1 THE INTEGERSwhere a an
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26 CHAPTER 1 THE INTEGERSEvery good
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28 CHAPTER 1 THE INTEGERSThe Euclid
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30 CHAPTER 1 THE INTEGERSTheorem 1.
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32 CHAPTER 1 THE INTEGERS9. Use ind
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34 CHAPTER 1 THE INTEGERSProgrammin
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36 CHAPTER 2 GROUPSZ n . Consider t
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38 CHAPTER 2 GROUPSSymmetriesA symm
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40 CHAPTER 2 GROUPSTable 2.2. Symme
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42 CHAPTER 2 GROUPSand 4. We define
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44 CHAPTER 2 GROUPSis the inverse o
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46 CHAPTER 2 GROUPS1. mg + ng = (m
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48 CHAPTER 2 GROUPSnecessarily obta
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50 CHAPTER 2 GROUPS3. Write out Cay
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52 CHAPTER 2 GROUPS32. Find all the
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54 CHAPTER 2 GROUPS0 50000 30042 6F
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3Cyclic GroupsThe groups Z and Z n
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58 CHAPTER 3 CYCLIC GROUPS✦ ❛
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60 CHAPTER 3 CYCLIC GROUPS3.2 The G
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62 CHAPTER 3 CYCLIC GROUPSanda = r
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64 CHAPTER 3 CYCLIC GROUPSTheorem 3
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66 CHAPTER 3 CYCLIC GROUPSk multipl
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68 CHAPTER 3 CYCLIC GROUPS4. Find t
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70 CHAPTER 3 CYCLIC GROUPS27. If g
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4Permutation GroupsPermutation grou
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74 CHAPTER 4 PERMUTATION GROUPSthis
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76 CHAPTER 4 PERMUTATION GROUPSExam
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78 CHAPTER 4 PERMUTATION GROUPSExam
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80 CHAPTER 4 PERMUTATION GROUPSProo
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82 CHAPTER 4 PERMUTATION GROUPSresu
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84 CHAPTER 4 PERMUTATION GROUPSand
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86 CHAPTER 4 PERMUTATION GROUPS(a)
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88 CHAPTER 4 PERMUTATION GROUPS(b)
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90 CHAPTER 5 COSETS AND LAGRANGE’
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98 CHAPTER 6 INTRODUCTION TO CRYPTO
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100 CHAPTER 6 INTRODUCTION TO CRYPT
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7Algebraic Coding TheoryCoding theo
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110 CHAPTER 7 ALGEBRAIC CODING THEO
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Page 146 and 147: 8IsomorphismsMany groups may appear
Page 148 and 149: 140 CHAPTER 8 ISOMORPHISMSab ≠ ba
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Page 154 and 155: 146 CHAPTER 8 ISOMORPHISMSTherefore
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Page 158 and 159: 150 CHAPTER 8 ISOMORPHISMS20. Prove
Page 160 and 161: 9Homomorphisms and FactorGroupsIf H
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Page 178 and 179: 10Matrix Groups andSymmetryWhen Fel
Page 180 and 181: 172 CHAPTER 10 MATRIX GROUPS AND SY
Page 198 and 199: 11The Structure of GroupsThe ultima
Page 200 and 201: 192 CHAPTER 11 THE STRUCTURE OF GRO
Page 212 and 213: 204 CHAPTER 12 GROUP ACTIONSThe ele
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Page 216 and 217: 208 CHAPTER 12 GROUP ACTIONSExample
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Page 220 and 221: 212 CHAPTER 12 GROUP ACTIONSinduces
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Page 228 and 229: 13The Sylow TheoremsWe already know
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Page 240 and 241: 14RingsUp to this point we have stu
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238 CHAPTER 14 RINGSProposition 14.
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240 CHAPTER 14 RINGSa ring homomorp
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242 CHAPTER 14 RINGSTheorem 14.9 Le
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244 CHAPTER 14 RINGSTheorem 14.15 L
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246 CHAPTER 14 RINGSthe Senate is n
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248 CHAPTER 14 RINGShas a solution.
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250 CHAPTER 14 RINGSMultiplying the
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252 CHAPTER 14 RINGS11. Prove that
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254 CHAPTER 14 RINGS34. Let R be a
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15PolynomialsMost people are fairly
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258 CHAPTER 15 POLYNOMIALSExample 1
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260 CHAPTER 15 POLYNOMIALSProof. Su
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262 CHAPTER 15 POLYNOMIALSm ≥ n.
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264 CHAPTER 15 POLYNOMIALSPropositi
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268 CHAPTER 15 POLYNOMIALSwhere eac
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272 CHAPTER 15 POLYNOMIALS(c) p(x)
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274 CHAPTER 15 POLYNOMIALSAdditiona
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276 CHAPTER 15 POLYNOMIALS(a) x 4
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278 CHAPTER 16 INTEGRAL DOMAINSelem
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280 CHAPTER 16 INTEGRAL DOMAINSIt i
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282 CHAPTER 16 INTEGRAL DOMAINSLet
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284 CHAPTER 16 INTEGRAL DOMAINSCons
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286 CHAPTER 16 INTEGRAL DOMAINSEucl
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288 CHAPTER 16 INTEGRAL DOMAINSin D
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290 CHAPTER 16 INTEGRAL DOMAINSCoro
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292 CHAPTER 16 INTEGRAL DOMAINS7. L
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17Lattices and BooleanAlgebrasThe a
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296 CHAPTER 17 LATTICES AND BOOLEAN
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18Vector SpacesIn a physical system
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314 CHAPTER 18 VECTOR SPACES5. −(
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316 CHAPTER 18 VECTOR SPACESProposi
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318 CHAPTER 18 VECTOR SPACES3. If S
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320 CHAPTER 18 VECTOR SPACES(d) Let
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19FieldsIt is natural to ask whethe
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324 CHAPTER 19 FIELDS· 0 1 α 1 +
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326 CHAPTER 19 FIELDSExample 5. We
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328 CHAPTER 19 FIELDSfor b i and c
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330 CHAPTER 19 FIELDSLetu = ∑ i,j
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332 CHAPTER 19 FIELDSwhere each fie
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334 CHAPTER 19 FIELDSLet F be a fie
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336 CHAPTER 19 FIELDSTheorem 19.19
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338 CHAPTER 19 FIELDSlength αβ. A
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340 CHAPTER 19 FIELDSbe the equatio
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342 CHAPTER 19 FIELDScosine,4α 3
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344 CHAPTER 19 FIELDS11. Let p(x) b
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20Finite FieldsFinite fields appear
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348 CHAPTER 20 FINITE FIELDSextensi
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350 CHAPTER 20 FINITE FIELDSGF(p 24
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352 CHAPTER 20 FINITE FIELDSMessage
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354 CHAPTER 20 FINITE FIELDSThe add
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356 CHAPTER 20 FINITE FIELDSExample
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358 CHAPTER 20 FINITE FIELDSand g(x
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360 CHAPTER 20 FINITE FIELDS(3) ⇒
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362 CHAPTER 20 FINITE FIELDS22. Sho
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21Galois TheoryA classic problem of
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366 CHAPTER 21 GALOIS THEORYthe Gal
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368 CHAPTER 21 GALOIS THEORYof G(E/
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370 CHAPTER 21 GALOIS THEORYin K. A
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372 CHAPTER 21 GALOIS THEORYThe pro
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374 CHAPTER 21 GALOIS THEORY{id, σ
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376 CHAPTER 21 GALOIS THEORYThe ker
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378 CHAPTER 21 GALOIS THEORYfor res
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380 CHAPTER 21 GALOIS THEORYis a no
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382 CHAPTER 21 GALOIS THEORYand onc
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384 CHAPTER 21 GALOIS THEORY(a) x 5
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386 CHAPTER 21 GALOIS THEORY[3] Fra
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388 NOTATIONSymbol Description Page
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390 NOTATIONSymbol Description Page
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392 HINTS AND SOLUTIONSConversely,
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394 HINTS AND SOLUTIONS4. (a)(c)( 1
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396 HINTS AND SOLUTIONS7. (a) (0000
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398 HINTS AND SOLUTIONS9. For any h
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400 HINTS AND SOLUTIONS2. The four
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402 HINTS AND SOLUTIONSChapter 17.
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404 HINTS AND SOLUTIONSChapter 19.
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GNU Free Documentation LicenseVersi
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408 GFDL LICENSEappear in the title
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410 GFDL LICENSEH. Include an unalt
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412 GFDL LICENSEply to the other wo
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IndexG-equivalent, 205G-set, 203nth
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416 INDEXEquivalence relation, 15Eu
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418 INDEXKlein, Felix, 46, 170, 245
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420 INDEXnormalizer of, 222proper,
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