Advanced High-School Mathematics

SECTION 4.2 Basics of Group Theory 211 Definition of Binary Operation on a Set. Abinary operation on a non-empty setSis a mappin ...

212 CHAPTER 4 Abstract Algebra Sinceτ is one-to-one, we conclude that s=s′, which proves thatσ◦τ is also one-to-one. σ◦τ is onto ...

SECTION 4.2 Basics of Group Theory 213 Note that the above example depends heavily on the fact thatZ is closed under both additi ...

214 CHAPTER 4 Abstract Algebra We display addition and multiplication on the integers modulo 5 in the following obvious tables: ...

SECTION 4.2 Basics of Group Theory 215 Are the non-zero elements{ 1 , 2 , 3 , 4 , 5 }inZ 6 closed under mul- tiplication? Are t ...

216 CHAPTER 4 Abstract Algebra If the binary operation has an identitye, then this identity isunique. Indeed, ife′were another i ...

SECTION 4.2 Basics of Group Theory 217 LetAbe a nonempty set. For any non-empty setB⊆Afind the inverse ofBwith respect to symme ...

218 CHAPTER 4 Abstract Algebra We have already noted on page 216 that the identity element and inverses are unique. This says th ...

SECTION 4.2 Basics of Group Theory 219 Furthermore, for allσ∈Sym(X), and for allx∈X, we have e◦σ(x) = e(σ(x)) = σ(x), and σ◦e(x) ...

220 CHAPTER 4 Abstract Algebra on page 208, we see that Gis closed under ◦. Since we already know that◦is associative on Sym(X) ...

SECTION 4.2 Basics of Group Theory 221 If any two of these elements are the same, say a′a = a′′a, for distinct elementsa′anda′′, ...

222 CHAPTER 4 Abstract Algebra subtle than it looks, and shall be the topic of the next subsection. A group (G,∗) is calledAbeli ...

SECTION 4.2 Basics of Group Theory 223 (i) Show that the six elementse, σ, σ^2 , τ, στ, σ^2 τ comprise all of the elements of th ...

224 CHAPTER 4 Abstract Algebra (c) Show that for every element x ∈ U, x^3 = e, where e is the identity ofU. (d) Show thatU isnot ...

SECTION 4.2 Basics of Group Theory 225 proving thatGis abelian. Let’s look at a few examples. The infinite additive group (Z,+) ...

226 CHAPTER 4 Abstract Algebra Z∗p for infinitely many primes p. This is often called theArtin Conjecture, and the answer is “ye ...

SECTION 4.2 Basics of Group Theory 227 o(1) = 1, o(2) = 3, o(3) = 6, o(4) = 3, o(5) = 6, o(6) = 2. Note that if the elementghas ...

228 CHAPTER 4 Abstract Algebra Let G be a finite group of order n, and assume that G has a elementgof ordern. Show thatGis a cy ...

SECTION 4.2 Basics of Group Theory 229 (b) If|H|<∞andHis closed under∗, thenH is a subgroup ofG. Proof. Notice first that we ...

230 CHAPTER 4 Abstract Algebra Show that the even integers 2Zis a subgroup of the additive group of the integers (Z,+). In fact ...