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On the Connections Between Set Theory and Group Theory

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On the Connections Between Set Theory and Group Theory ON THE CONNECTIONS BETWEEN SET THEORY AND GROUP THEORY DINGDING DONG Abstract. This paper is an introduction to the connections between set the- ory and group theory. After introducing the notion of cardinal arithmetic, we prove the Schroeder-Bernstein Theorem about cardinal equivalence. We then discuss how sets and groups exhibit the properties of each other, specifically, the cardinal relations between groups and the group structures within sets of different cardinalities. Contents 1. Introduction 1 2. Cardinal Arithmetic 1 3. Schroeder-Bernstein Theorem 4 4. Lagrange's Theorem 6 5. Cyclic Groups 7 6. Understanding Sets within Group Structures 8 Acknowledgments 10 References 10 1. Introduction The exploration of uncountable sets dates back to the nineteenth century, when Georg Cantor proved the existence of infinite sets that cannot be put into one- to-one correspondence with the set of natural numbers. Through the diagonal argument, Cantor showed that there is no bijection between the sets of real and natural numbers. Stemmed from this discovery, the notion of cardinality helped mathematicians understand more about sets that seem not as approachable. Groups are a particular kind of sets equipped with a binary operation, which satisfies the four group axioms of closure, associativity, identity and invertibility. Since groups are still sets, anything about sets still holds true in groups. However, due to their specific properties, there is more accessible information about groups themselves. 2. Cardinal Arithmetic Cardinal numbers measure the cardinality, or size of sets. In terms of bijective functions, it defines an equivalence relation between sets. Definition 2.1. Two sets A and B are of the same cardinality if there exists a bijection f : A ! B between A and B. 1 2 DINGDING DONG Definition 2.2. A cardinal number is thought of as an equivalence class of cardi- nality, i.e. card(A) = card(B) if A and B are of the same cardinality. Definition 2.3. Let a; b be cardinal numbers. If there exist sets A; B with card(A) = a, card(B) = b and A ⊆ B, then a ≤ b. Definition 2.4. Let a; b be cardinal numbers. If A; B are sets with card(A) = a, card(B) = b and A \ B = ?, then a + b = card(A [ B). Definition 2.5. Let a; b be cardinal numbers. If A; B are sets with card(A) = a and card(B) = b, then a · b = card(A × B). Definition 2.6. @0 = card(N). Theorem 2.7. For any infinite cardinal number a, @0 ≤ a. Proof. Let A be an infinite set with card(A) = a, and F0 ⊂ F1 ⊂ F2 ⊂ ::: be a sequence of subsets of A with jFij = i. Let xi 2 Fi n Fi−1. Then x1; x2; ::: 2 A with xi 6= xj for any i 6= j, fxi j i 2 Ng ⊆ A, card(fxi j i 2 Ng) ≤ card(A), @0 ≤ a. An important lemma in set theory is Zorn's Lemma, which identifies the existence of a maximal element within any partially ordered set that is bounded above. Definition 2.8. A partially ordered set is a set P with a binary relation ≤ such that, for all a; b; c 2 P : (1) (Reflexivity) a ≤ a. (2) (Antisymmetry) If a ≤ b and b ≤ a, then a = b. (3) (Transitivity) If a ≤ b and b ≤ c, then a ≤ c. Lemma 2.9. (Zorn's Lemma) Suppose a partially ordered set P has the property that every chain in P has an upper bound in P. Then the set P contains at least one maximal element. Theorem 2.10. Let a be an infinite cardinal number. Then a + a = a. Proof. Following the proof of Z. Nagy, let A be an infinite set with card(A) = a. Let S(A) be the set of families fAi j i 2 Ig of subsets of A with (a) card(Ai) = @0 for all i 2 I; (b) Ai \ Aj = ? for all i; j 2 I. S(A) with set inclusion is a partially ordered set. For any S 2 S(A), S ⊆ S. If S1;S2 2 S(A), S1 ⊆ S2 and S2 ⊆ S1, then S1 = S2. If S1;S2;S3 2 S(A), S1 ⊆ S2 and S2 ⊆ S3, then S1 ⊆ S3. Since S(A) with set inclusion is a partially ordered set and every chain in S(A) is bounded above by fAi j Ai ⊆ A; card(Ai) = @0g, S(A) contains a maximal element fAi j i 2 I0g. Let B = An(S A ). Since fA g is maximal, B is finite, card(A) = card(S A ). i2I0 i i i2I0 i Then a = card(A) = card(N × I0). Suppose card(I0) = c. Then a = @0 · c. Let C0 = fn 2 N j n eveng, C1 = fn 2 N j n oddg. Since there are bijections f(x) = 2x between N and C0, g(x) = 2x − 1 between N and C1, card(C0) = card(C1) = @0. Since (C0 × I0) [ (C1 × I0) = N × I0,(C0 × I0) \ (C1 × I0) = ?, ON THE CONNECTIONS BETWEEN SET THEORY AND GROUP THEORY 3 a = card(N × I0) = card(C0 × I0) + card(C1 × I0) = @0 · card(I0) + @0 · card(I0) = @0 · c + @0 · c = a + a: Theorem 2.11. If a; b are cardinal numbers with 1 ≤ b ≤ a and a infinite, then a · b = a. Proof. Since a ≤ a · b ≤ a · a, it is sufficient to prove that a = a · a. Define χ = f(D; f): D ⊂ A; f : D ! D × D bijectiveg; and equip χ with the order ( 0 0 0 D ⊂ D (D; f) ≺ (D ; f ) () 0 : f = f jD S Let τ = f(Di; fi): i 2 Ig be a totally ordered subset of χ, D0 = i2I Di, f : D0 ! D0 × D0 be the unique function with fjDi = fi. Then f is injective with [ [ [ f(D0) = f(Di) = fi(Di) = (Di × Di) = D0 × D0: i2I i2I i2I (D; f) is an upper bound of τ. According to Zorn's Lemma, since 8τ ⊆ χ is bounded above, there exists a maximal element (D; f) 2 χ. Let card(D) = d. Since (D; f) 2 χ, d · d = d. If d = a, then a · a = a. By contradiction, assume that d 6= a. Since d ≤ a as D ⊂ A, d < a. Let G = A n D. Then d + card(G) = a. Since d < a, card(G) = a by Theorem 2.10, and there exists a subset E ⊂ G with card(E) = d. Consider the set P = (E × E) [ (E × D) [ (D × E): Since E \D = ?,(E ×E); (E ×D); (D×E) are disjoint from each other. According to Theorem 2.10, card(P ) = card(E × E) + card(E × D) + card(D × E) = d · d + d · d + d · d = d = card(E): There exists a bijection g : E ! P . Since E \ D = P \ (D × D) = ?, there exists a bijection h : D [ E ! P [ (D × D) such that hjD = f and hjE = g. Since P [(D ×D) = (D [E)×(D [E), (D [E; h) 2 χ,(D; f) is not the maximal element in χ. )( Hence d = a, a · a = a. 4 DINGDING DONG 3. Schroeder-Bernstein Theorem Schroeder-Bernstein Theorem states that if there exists an injection from set A to set B and an injection from set B to set A, then there exists a bijection between A and B. In terms of cardinality, it means that if jAj ≤ jBj and jBj ≤ jAj, then jAj = jBj, which completes the ordering of cardinal numbers. Proposition 3.1. If B ⊆ A and there exists an injective function f : A ! B, then there exists a bijective function h : A ! B between A and B. Proof. If X ⊆ Y , then ff(k) j k 2 Xg ⊆ ff(k) j k 2 Y g, f(X) ⊆ f(Y ). Since in the base case, we have A ⊇ B, B ⊇ f(A), A ⊇ B ⊇ f(A) ⊇ f(B) ⊇ f 2(A) ⊇ f 2(B) ⊇ ::: by induction. Consider the sequence of sets X0 = A n B; X1 = B n f(A);X2 = f(A) n f(B); ::: Since for any n, Xn+2 = f(P ) n f(Q) while Xn = P n Q, for any x 2 Xn+2, x 2 f(P ), there exists p 2 P such that f(p) = x. Since p2 = Q (or x = f(p) 2 f(Q)), p 2 P n Q = Xn with f(p) = x. Hence f is both surjective and injective, there exists a bijection f : Xn ! Xn+2 between Xn and Xn+2. For any n; m 2 N such that n > m, since Xm \ Xm+1 = ? and Xm+1 ⊇ Xn, Xm \ Xn = ?. Hence for sets 1 1 [ [ V = Xn;W = Xn; n=0 n=1 there exists a bijection F : V ! W between V and W with ( x if x 2 X ; n odd F (x) = n : f(x) if x 2 Xn; n even Since B ⊆ A, V ⊆ A, W ⊆ V , W ⊆ B, A = (A n (B [ V )) [ (B n V ) [ (V n B) [ ((B \ V ) n W ) [ W B = (B n V ) [ ((B \ V ) n W ) [ W V = (V n B) [ ((B \ V ) n W ) [ W with (A n (B [ V )); (B n V ); (V n B); ((B \ V ) n W );W disjoint from each other. Since A n B = X0 = V n W , (A n (B [ V )) [ (V n B) = A n B = V n W = (V n B) [ ((B \ V ) n W ) (A n (B [ V )) = ((B \ V ) n W ) A n V = (A n (B [ V )) [ (B n V ) = (B n V ) [ ((B \ V ) n W ) = B n W Let Z = A n V = B n W .
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