Download e-book for kindle: Presentations of Groups by D. L. Johnson

By D. L. Johnson

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Thus, k ≡ 0 (mod 5), so σ xi = x h . Therefore, if σ X i = X h , then, for any integer k, σρ m xi = ρ m x h . 22) Similarly, if σ (B j ) = Bh , then, for any integer m, σρ m B j = ρ m Bh . 23). Let Y = {y ∈ X : σ y = y}. Since each σ -orbit on X is of cardinality 1 or 3, we obtain that |Y | ≡ 41 ≡ 2 (mod 3). Since σ fixes ∞ and either all or none of the points of each ρ -orbit on X , we obtain that |Y | ≡ 1 (mod 5). , |Y | is 11 or 26. We claim that |Y | = 11. Suppose |Y | = 26. 9, the set C of fixed blocks of σ is of cardinality 26.

X 8 } is of cardinality 1 or 3 and since σ fixes only two elements of this set, we obtain that σ cyclically permutes X 1 , X 2 , and X 3 . Therefore, we assume without loss of generality, that σ acts on the set {X 1 , X 2 , . . , X 8 } as the permutation (X 1 X 2 X 3 )(X 4 X 5 X 6 )(X 7 )(X 8 ). Let Y1 = X 1 ∪ X 2 ∪ X 3 , Y2 = X 4 ∪ X 5 ∪ X 6 , and Y3 = X 7 ∪ X 8 . Similarly, we assume that σ acts on the set {B1 , B2 , . . , B8 } as the permutation (B1 B2 B3 )(B4 B5 B6 )(B7 )(B8 ). We have now described the action of ρ, τ , and σ on both X and B.

Suppose further that there exists a nonnegative integer λ such that (v − 1)λ = r (k − 1) and (i) any two points of D are incident with at most λ blocks or (ii) any two points of D are incident with at least λ blocks. Then D is a (v, b, r, k, λ)-design. Introduction to designs 28 Proof. Fixing a point x ∈ X and counting flags (y, B) where x is incident with B yields either (v − 1)λ ≥ r (k − 1) or (v − 1)λ ≤ r (k − 1), respectively. Since, in fact, (v − 1)λ = r (k − 1), we obtain that in either case there are exactly λ blocks containing {x, y}.

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