Expanding Generator Sets for Solvable Permutation Groups V. Arvind † Partha Mukhopadhyay ∗ Prajakta Nimbhorkar ∗ Yadu Vasudev † November 25, 2011 Abstract Let G = 〈S 〉 be a solvable permutation group given as input by the generating set S . I.e. G is a solvable subgroup of the symmetric group S n . We give a deterministic polynomial-time algorithm that computes an expanding generating set of size  O(n 2 ) for G. More precisely, given a λ< 1, we can compute a subset T ⊂ G of size  O(n 2 ) ( 1 λ ) O(1) such that the undirected Cayley graph Cay(G, T ) is a λ-spectral expander (the  O notation suppresses log O(1) n factors). In particular, this construction yields ε-bias spaces with improved size bounds for the groups Z n d for any constant ε> 0. We also note that for any permutation group G ≤ S n given by a generating set, in deterministic polynomial time we can compute an ( n λ ) O(1) size expanding generating set T , such that Cay(G, T ) is a λ-spectral expander; here the constant in the exponent is large but independent of λ. 1 Introduction Let G be a finite group, and let S = 〈g 1 ,g 2 ,...,g k 〉 be a generating set for G. The undirected Cayley graph Cay(G, S ∪ S -1 ) is an undirected multigraph with vertex set G and edges of the form {x, xg i } for each x ∈ G and g i ∈ S . Since S is a generating set for G, Cay(G, S ∪ S -1 ) is a connected regular multigraph. For a regular undirected graph X =(V,E) of degree D on n vertices, its normalized adjacency matrix A X is a symmetric matrix with largest eigenvalue 1. For 0 <λ< 1, the graph X is an (n, D, λ)-spectral expander if the second largest eigenvalue of A X , in absolute value, is bounded by λ. Expander graphs are of great interest and importance in theoretical computer science, especially in the study of randomness in computation; the monograph by Hoory, Linial, and * Chennai Mathematical Institute, Siruseri, India. Emails: {partham,prajakta}@cmi.ac.in † The Institute of Mathematical Sciences, Chennai, India.Emails: {arvind,yadu}@imsc.res.in 1 ISSN 1433-8092