Advanced High-School Mathematics

SECTION 4.2 Basics of Group Theory 231 (a) Show that GL 2 (Zp) is a group. (b) Show that there arep(p+ 1)(p−1)^2 elements in thi ...

232 CHAPTER 4 Abstract Algebra reflexivity: g≡ g( modH) sinceg−^1 g=e∈H. symmetry: If g ≡ g′( modH) then g−^1 g′ ∈ H, and so g′− ...

SECTION 4.2 Basics of Group Theory 233 which has cardinality |H|. If G is partitioned into k such sets, then obviously|G|=k|H|, ...

234 CHAPTER 4 Abstract Algebra see that|G| ∣∣ ∣24. But there are plenty of permutations of the vertices 1, 2, 3, and 4 which are ...

SECTION 4.2 Basics of Group Theory 235 LetGbe a group, letHbe a subgroup, and recall the equivalence relation ( modH) defined b ...

236 CHAPTER 4 Abstract Algebra Here’s a much less obvious example. Consider the two infinite groups (R,+) and (R+,·). At first b ...

SECTION 4.2 Basics of Group Theory 237 bijective. Note that in this case, the inverse mappingf−^1 :H→Gis also a homomorphism. Th ...

238 CHAPTER 4 Abstract Algebra that f is also one-to-one. (Is this obvious?) Finally, let xk, xl ∈ G; ifk+l ≤n−1, thenf(xkxl) =f ...

SECTION 4.2 Basics of Group Theory 239 a homomorphism of GL 2 (R) into the multiplicative group of non- zero real numbers. (Rea ...

240 CHAPTER 4 Abstract Algebra Let R be the additive group of real numbers and assume that f :R→Ris a function which satisfies ...

SECTION 4.2 Basics of Group Theory 241 for any two vertices, call themx 1 andx 2 , there is always an automor- phismσwhich carri ...

242 CHAPTER 4 Abstract Algebra σ≡σ′(modH) ⇐⇒σ−^1 σ′∈H. Recall also from Subsection 4.2.7 that the equivalence classes each have ...

SECTION 4.2 Basics of Group Theory 243 IfHis the stabilizer of the vertex 1, then surely Hmust permute the three vertices 2, 4, ...

244 CHAPTER 4 Abstract Algebra From the above we conclude that the automorphism group of this graph has 6×2 = 12 elements, meani ...

Chapter 5 Series and Differential Equations The methods and results of this chapter pave the road to the students’ more serious ...

246 CHAPTER 5 Series and Differential Equations |f(x)−L|< . Notice that in the above definition we stipulate 0<|x−a|< ...

SECTION 5.1 Quick Survey of Limits 247 Definition. Let f be a function defined on an interval of the form a < x < b. We sa ...

248 CHAPTER 5 Series and Differential Equations Next, assume that 1>  >0, and letδ > 0 be a real number such that when ...

SECTION 5.1 Quick Survey of Limits 249 Definition. Letf be a function defined in a neighborhood ofa. If limx→a f(x)−f(a) x−a =L, ...

250 CHAPTER 5 Series and Differential Equations U(f;P) = ∑n i=i Mi(xi−xi− 1 ), and the lower Riemann sum relative to the above p ...