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402 HINTS AND SOLUTIONSChapter 17. Lattices and Boolean Algebras2.5. False.6. (a) (a ∨ b ∨ a ′ ) ∧ a.30☞ ❅☞ ❅☞ 10 15☞☞ 2 5 3❅❅ 1abaa ′
HINTS AND SOLUTIONS 403(c) a ∨ (a ∧ b).ab8. Not equivalent.10. a ′ ∧ [(a ∧ b ′ ) ∨ b] = a ∧ (a ∨ b).15. Let I, J be ideals in R. We need to show that I+J = {r+s : r ∈ I and s ∈ J}is the smallest ideal in R containing both I and J. If r 1 , r 2 ∈ I and s 1 , s 2 ∈ J,then (r 1 + s 1 ) + (r 2 + s 2 ) = (r 1 + r 2 ) + (s 1 + s 2 ) is in I + J. For a ∈ R,a(r 1 + s 1 ) = ar 1 + as 1 ∈ I + J; hence, I + J is an ideal in R.19. (a) No.21. (⇒). a = b ⇒ (a ∧ b ′ ) ∨ (a ′ ∧ b) = (a ∧ a ′ ) ∨ (a ′ ∧ a) = O ∨ O = O.(⇐). (a∧b ′ )∨(a ′ ∧b) = O ⇒ a∨b = (a∨a)∨b = a∨(a∨b) = a∨[I ∧(a∨b)] =a∨[(a∨a ′ )∧(a∨b)] = [a∨(a∧b ′ )]∨[a∨(a ′ ∧b)] = a∨[(a∧b ′ )∨(a ′ ∧b)] = a∨0 = a.A symmetric argument shows that a ∨ b = b.Chapter 18. Vector Spaces3. Q( √ 2, √ 3 ) has basis {1, √ 2, √ 3, √ 6 } over Q.5. P n has basis {1, x, x 2 , . . . , x n−1 }.7. (a) Subspace of dimension 2 with basis {(1, 0, −3), (0, 1, 2)}.(d) Not a subspace.10. 0 = α0 = α(−v + v) = α(−v) + αv ⇒ −αv = α(−v).12. Let v 0 = 0, v 1 , . . . , v n ∈ V and α 0 ≠ 0, α 1 , . . . , α n ∈ F . Then α 0 v 0 + · · · +α n v n = 0.15. (a) Let u, v ∈ ker(T ) and α ∈ F . ThenaT (u + v) = T (u) + T (v) = 0T (αv) = αT (v) = α0 = 0.Hence, u + v, αv ∈ ker(T ) ⇒ ker(T ) is a subspace of V .(c) T (u) = T (v) ⇔ T (u − v) = T (u) − T (v) = 0 ⇔ u − v = 0 ⇔ u = v.17. (a) Let u, u ′ ∈ U and v, v ′ ∈ V . Then(u + v) + (u ′ + v ′ ) = (u + u ′ ) + (v + v ′ ) ∈ U + Vα(u + v) = αu + αv ∈ U + V.
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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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8IsomorphismsMany groups may appear
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140 CHAPTER 8 ISOMORPHISMSab ≠ ba
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142 CHAPTER 8 ISOMORPHISMSThe addit
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144 CHAPTER 8 ISOMORPHISMSthat is,
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146 CHAPTER 8 ISOMORPHISMSTherefore
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148 CHAPTER 8 ISOMORPHISMSWe will l
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150 CHAPTER 8 ISOMORPHISMS20. Prove
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9Homomorphisms and FactorGroupsIf H
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154 CHAPTER 9 HOMOMORPHISMS AND FAC
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10Matrix Groups andSymmetryWhen Fel
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172 CHAPTER 10 MATRIX GROUPS AND SY
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11The Structure of GroupsThe ultima
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192 CHAPTER 11 THE STRUCTURE OF GRO
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204 CHAPTER 12 GROUP ACTIONSThe ele
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206 CHAPTER 12 GROUP ACTIONSand the
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208 CHAPTER 12 GROUP ACTIONSExample
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210 CHAPTER 12 GROUP ACTIONSLemma 1
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212 CHAPTER 12 GROUP ACTIONSinduces
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214 CHAPTER 12 GROUP ACTIONSSwitchi
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216 CHAPTER 12 GROUP ACTIONSTable 1
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218 CHAPTER 12 GROUP ACTIONS10. Fin
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13The Sylow TheoremsWe already know
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222 CHAPTER 13 THE SYLOW THEOREMSHe
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224 CHAPTER 13 THE SYLOW THEOREMSpo
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226 CHAPTER 13 THE SYLOW THEOREMSbe
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228 CHAPTER 13 THE SYLOW THEOREMSSu
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230 CHAPTER 13 THE SYLOW THEOREMS(e
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14RingsUp to this point we have stu
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234 CHAPTER 14 RINGSExample 3. We c
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236 CHAPTER 14 RINGS3. (−a)(−b)
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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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Page 372 and 373: 21Galois TheoryA classic problem of
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Page 396 and 397: 388 NOTATIONSymbol Description Page
Page 398 and 399: 390 NOTATIONSymbol Description Page
Page 400 and 401: 392 HINTS AND SOLUTIONSConversely,
Page 402 and 403: 394 HINTS AND SOLUTIONS4. (a)(c)( 1
Page 404 and 405: 396 HINTS AND SOLUTIONS7. (a) (0000
Page 406 and 407: 398 HINTS AND SOLUTIONS9. For any h
Page 408 and 409: 400 HINTS AND SOLUTIONS2. The four
Page 412 and 413: 404 HINTS AND SOLUTIONSChapter 19.
Page 414 and 415: GNU Free Documentation LicenseVersi
Page 416 and 417: 408 GFDL LICENSEappear in the title
Page 418 and 419: 410 GFDL LICENSEH. Include an unalt
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Page 422 and 423: IndexG-equivalent, 205G-set, 203nth
Page 424 and 425: 416 INDEXEquivalence relation, 15Eu
Page 426 and 427: 418 INDEXKlein, Felix, 46, 170, 245
Page 428: 420 INDEXnormalizer of, 222proper,
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