Growth and generation in SL2(Z/pZ
نویسنده
چکیده
We show that every subset of SL2(Z/pZ) grows rapidly when it acts on itself by the group operation. It follows readily that, for every set of generators A of SL2(Z/pZ), every element of SL2(Z/pZ) can be expressed as a product of at most O((log p)) elements of A ∪ A, where c and the implied constant are absolute.
منابع مشابه
Growth in Groups: Ideas and Perspectives
This is a survey of methods developed in the last few years to prove results on growth in non-commutative groups. These techniques have their roots in both additive combinatorics and group theory, as well as other fields. We discuss linear algebraic groups, with SL2(Z/pZ) as the basic example, as well as permutation groups. The emphasis will lie on the ideas behind the methods.
متن کاملGrowth in SL3(Z/pZ)
Let G = SL3(Z/pZ), p a prime. Let A be a set of generators of G. Then A grows under the group operation. To be precise: denote by |S| the number of elements of a finite set S. Assume |A| < |G| for some ǫ > 0. Then |A ·A ·A| > |A| , where δ > 0 depends only on ǫ. We will also study subsets A ⊂ G that do not generate G. Other results on growth and generation follow.
متن کاملAn algorithm for the Baker-Campbell-Hausdorff formula
A simple algorithm, which exploits the associativity of the BCH formula, and that can be generalized by iteration, extends the remarkable simplification of the BakerCampbell-Hausdorff (BCH) formula, recently derived by Van-Brunt and Visser. We show that if [X,Y ] = uX + vY + cI, [Y, Z] = wY + zZ + dI, and, consistently with the Jacobi identity, [X,Z] = mX + nY + pZ + eI, then exp(X) exp(Y ) exp...
متن کاملExplicit Growth and Expansion for Sl2
We give explicit versions of Helfgott’s Growth Theorem for SL2, as well as of the Bourgain-Gamburd argument for expansion of Cayley graphs modulo primes of subgroups of SL2(Z) which are Zariski-dense in SL2.
متن کاملA simulation study of RX-mode waves generation in the equatorial plasmasphere
The generation mechanism of RX-mode waves in the equatorial plasmasphere has not been well understood. The Akebono passing through the storm time geomagnetic equator shows the possibility of the local enhancement of RX-mode waves in association with intense Z-mode waves in the equatorial region. We use the initial parameters inferred from observational data from around the plasma-wave generatio...
متن کامل