Alice C. Niemeyer

A black-box group algorithm for recognizing finite symmetric and alternating groups, I

We present a Las Vegas algorithm which, for a given black-box group known to be isomorphic to a symmetric or alternating group, produces an explicit isomorphism with the standard permutation representation of the group. This algorithm has applications in computations with matrix groups and permutation groups. In this paper, we handle the case when the degree n of the standard permutation representation is part of the input. In a sequel, we shall treat the case when the value of n is not known in advance. As an important ingredient in the theoretical basis for the algorithm, we prove the following result about the orders of elements of S n : the conditional probability that a random element σ ∈ S n is an n-cycle, given that σ n = 1, is at least 1/10.

Permutations with Restricted Cycle Structure and an Algorithmic Application

Let q be an integer with q ≥ 2. We give a new proof of a result of Erdos and Turan determining the proportion of elements of the finite symmetric group Sn having no cycle of length a multiple of q. We then extend our methods to the more difficult case of obtaining the proportion of such elements in the finite alternating group An. In both cases, we derive an asymptotic formula with error term for the above mentioned proportion, which contains an unexpected occurrence of the Gamma-function.We apply these results to estimate the proportion of elements of order 2f in Sn, and of order 3f in An and Sn, where gcd(2, f) = 1, and gcd(3, f) = 1, respectively, and log f is polylogarithmic in n. We also give estimates for the probability that the fth power of such elements is a transposition or a 3-cycle, respectively. An algorithmic application of these results to computing in An or Sn, given as a black-box group with an order oracle, is discussed.

A Finite Soluble Quotient Algorithm

An algorithm for computing power conjugate presentations for finite soluble quotients of predetermined structure of finitely presented groups is described. Practical aspects of an implementation are discussed.

The block-transitive, point-imprimitive 2-(729, 8, 1) designs

AbstractThe block-transitive point-imprimitive 2-(729,8,1) designs are classified. They all have full automorphism group of order 729.13 which is an extension of a groupN of order 729, acting regularly on points, by a group of order 13. There are, up to isomorphism, 27 designs withN elementary abelian, 13 designs withN=Z93 and 427 designs withN the relatively free 3-generator, exponent 3, nilpotency class 2 group, a total of 467 designs. This classification completes the classification of block-transitive, point-imprimitive 2-(ν, k, 1) designs satisfying

Abundant p-singular elements in finite classical groups

\upsilon = \left( {\left( {_2^k } \right) - 1} \right)^2

Groups with exponent six

, which is the Delandtsheer-Doyen upper bound for the numberν of points of such designs. The only examples of block-transitive, point-imprimitive 2-(ν, k, 1) designs with

The minimum length of a base for the symmetric group acting on partitions

\upsilon = \left( {\left( {_2^k } \right) - 1} \right)^2