Three problems in the asymptotic order of group elements

James Thomas

doi:10.17918/00000255

This thesis investigates three distinct questions concerning the asymptotic average order of certain group theoretic objects. The three main chapters can be read independently. In chapter zero, we give a high level overview of our results. For a polynomial f [is an element of] [finite field]q[x] with nonzero constant term, define the order of f to be the order of x in the quotient ring [finite field]q[x]/(f). In Chapter 1, we consider the average order of several classes of polynomials as the degree n [right arrow] [infinity]. We show that the average order of an irreducible polynomial is asymptotically [theta]([phi](q^n - 1)) with [phi] the Euler Totient function. Building on work of Stong [32], we show that the asymptotic average order of both squarefree and nonsquarefree polynomials is q^n / n^(1+o(1)). As an application we relate this to the expected orbit size of a vector under iteration by a random invertible matrix. In chapter two, we count the number of compositions of n with k parts whose pairwise gcds are coprime to a given finite set P of primes. We obtain asymptotic results when n [right arrow] [infinity], showing that the number of such compositions is ([product](p[is an element of]P|(n)) h[k](p) + (-1)^(k-1)(k-1)/p^(k-1) (1-1/p)) x ([product](p[is an element of P[crossed pipe](n)) h[k](p) + (-1)^k(k-1)/p^k) n^(k-1)/(k-1)! + O(n^(k-2)) where h[k](p) := (1-1/p)^(k-1)(1+(k-1)/p). Building on work of Bubbolini, Luca and Spiga [6], we also consider the set of compositions whose parts are pairwise coprime. We use a geometric argument to strengthen their result when k = 3. We apply their results to obtain bounds on the average order of a permutation with a given number of cycles. Chapter three is joint work with Huseyin Acan, Charles Burnette, Sean Eberhard, and Eric Schmutz. Motivated by a question of Thibault Godin [17], we show that the probability P₁ that two random permutations in the symmetric group S[n] have the same order is asymptotically n^(-2+o(1)). We compare this to the probability P₂ that two permutations are conjugate, showing by construction that lim sup(n) P₁/P₂ = [infinity]. Additionally, we show by more direct methods that with probability O(log log n / log n) there exists some prime dividing the order of one of the permutations but not the other.

Three problems in the asymptotic order of group elements

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Three problems in the asymptotic order of group elements

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