On semiregular permutations of a finite set

Abstract

Abstract. In this paper we establish upper and lower bounds for the proportion of permutations in symmetric groups which power up to semiregular permutations (permutations all of whose cycles have the same length). Provided that an integer n has a divisor at most d, we show that the proportion of such elements in S n is at least cn −1+1/2d for some constant c depending only on d whereas the proportion of semiregular elements in S n is less than 2n −1 .

Fast recognition of alternating groups of unknown degree

“…Observe that t b (b) = 1/b, since the only allowable permutations are the b-cycles and the proportion of b-cycles in S b is 1/b. The proof of the following lemma refines the ideas in [10] to obtain the explicit lower bound given below.…”

“…The computational recognition of finite simple groups is a fundamental task in the finite matrix group recognition project (see [8,9,11]). Generally not much is known about the way in which a group might be given as input and therefore algorithms which take black-box groups (see [1]) as input are the most versatile.…”

Permutations with Orders Coprime to a Given Integer

“…Although m was assumed to be a prime in [8], the formula for p ¬m (n) in (1) holds for an arbitrary positive integer m, see [11], and their asymptotic arguments can be extended to give explicit convergence bounds [3,Theorem 2.3(b)], again for arbitrary m. These explicit bounds, together with analogous results for alternating groups [3,Section 3], were used to analyse algorithms for constructing transpositions and 3-cycles [3,Section 6], procedures used as components of the constructive recognition algorithms for black-box alternating and symmetric groups in [4]. Many other authors have also considered the proportion p ¬m (n), see for example [5,6,16] and the discussion in [17].…”