In a connected group of finite Morley rank in which, generically, elements belong to connected nilpotent subgroups, proper normalizing cosets of definable subgroups are not generous. We explain why this is true and what consequences this has on an abstract theory of Weyl groups in groups of finite Morley rank. The only known infinite simple groups of finite Morley rank are the simple algebraic ...

Let G be a group covered by its left cosets a1G1, · · · , akGk exactly m times. It is known that [G : ⋂k i=1 Gi] 6 k!. When all the Gi are subnormal in G and ⋂k i=1 Gi = H, we are able to determine the least value of k in terms of m,G,H. For any i = 1, · · · , k, providing G/(Gi)G is solvable we show that k > m + f([G : Gi]) and hence [G : Gi] 6 2k−m, where f(n) = ∑r s=1 αs(ps − 1) if p1 1 · · ...

Our goal will be to generalize the construction of the group Z/nZ. The idea there was to start with the group Z and the subgroup nZ = 〈n〉, where n ∈ N, and to construct a set Z/nZ which then turned out to be a group (under addition) as well. (There are two binary operations + and · on Z, but Z is just a group under addition. Thus, the fact that we can also define multiplication on Z/nZ will not...

We study the nonunitary diagonal cosets constructed from admissible representations of Kač-Moody algebras at fractional level, with an emphasis on the question of field identification. Generic classes of field identifications are obtained from the analysis of the modular S matrix. These include the usual class related to outer automorphisms, as well as some intrinsically nonunitary field identi...

Let g be a simple, finite-dimensional Lie (super)algebra, equipped with an embedding of sl2 inducing the minimal gradation on g. The corresponding minimal Walgebra W(g, e−θ) introduced by Kac and Wakimoto has strong generators in weights 1, 2, 3 2 , and all operator product expansions are known explicitly. The weight one subspace generates an affine vertex (super)algebra A(g) where g ⊂ g denote...

We study the spectra of G/G coset models by computing BRST cohomology of affine Lie algebras with coefficients in tensor product of two modules. One-to-one correspondence between the spectra of A1/A 1 1 and that of the minimal matter coupled to gravity (including boundary states of the Kac table) is observed. This phenomena is discussed from the point of hamiltonian reduction of BRST complexes ...

It is proved a generalization of Cusick-Cheon’s conjecture on balanced Boolean functions in the cosets of the binary Reed-Muller code RM (k ,m) of order k and length 2m, when k = 1 or k ≥ (m − 1)/2.

We study the question how many subgroups, cosets or subspaces are needed to cover a finite Abelian group or a vector space if we have some natural restrictions on the structure of the covering system. For example we determine, how many cosets we need, if we want to cover all but one element of an Abelian group. This result is a group theoretical extension of the theorem of Brouwer, Jamison and ...

We construct a large number of record-breaking binary, ternary and quaternary codes. Our methods involve the study of BCH-codes over larger fields, concatenation, construction X and variants of the Griesmer construction (residual codes). 1 Review of the theory Let IFq be the ground field, F = IFq2 . Denote the interval {i, i + 1, . . . , j} ⊂ ZZ/(q−1)ZZ by [i, j]. Let A = [i, j] ⊂ ZZ/(q−1)ZZ. I...

Let S be a closed, oriented surface of genus g ≥ 2, and consider the extension 1 → π1S → MCG(S, p) → MCG(S) → 1, where MCG(S) is the mapping class group of S, and MCG(S, p) is the mapping class group of S punctured at p. We prove that any quasi-isometry of MCG(S, p) which coarsely respects the cosets of the normal subgroup π1S is a bounded distance from the left action of some element of MCG(S,...