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By Eiichi Bannai

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Extra resources for Algebraic Combinatorics I: Association Schemes

Example text

Fk ) ∈ C(g1 , . . , gk ) Now we study the group G in a series of claims. Claim 2. G = G1 × G2 . Indeed, it suffices to show that G1 ∩ G2 = 1. 3 (h1 , . . , hk ) ∼ (g1 m1 , . . , gk−1 mk−1 , gk mk ) for some m1 , . . , mk ∈ M . 1, (g1 m1 , . . , gk−1 mk−1 , gk mk ) ∈ C(g1 , . . , gk ) gpG (g1 m1 ) = G1 , gpG (g2 m2 , . . , gk mk ) = G2 and we can represent the elements m2 , . . , mk ∈ G1 ∩G2 as products of conjugates of g1 m1 , therefore deducing that (g1 m1 , g2 m2 . . , gk mk ) ∼ (g1 m1 , g2 , .

Claim 3. Every two k-tuples U2 = (g, 1, . . , 1) and U3 = (h, 1, . . , 1) from Nk (G, Ω) are AC-equivalent. Indeed, U2 is AC-equivalent to (g, 1, . . , 1, g). 2 the former one is AC-equivalent to (h, . . , 1, g), which is AC-equivalent to (h, 1, . . , 1), as required. The theorem follows from Claims 1, 2, and 3. 6. 1. Let G = G1 × · · · × Gs × A (3) be a direct decomposition of an Ω-group G into a product of non-abelian Ω-simple Ω-groups Gi , i = 1, . . s, and an abelian Ω-group A. Then, assuming G = 1, dGΩ (G) = max{dAΩ (A), 1}.

Let (B(zi , η)1 i k be a finite cover of this compact set by closed balls of radius η, such that on each B(zi , η) the foliation is defined by a function Fi , and such that the balls B(zi , η/100) cover K. If L is a leaf of our foliation, and L pass through some y ∈ B(zi , η/100), the volume of the connected component of L ∩ B(zi , η/2) through y is α for some universal constant α . From the bound on the volume of [Lxn ], we deduce that there exists a uniform bound on the number of connected component of Lxn throughB(zi , η/100).

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