Rewriteability in Finite Groups

What\u27s the probability that two elements in a finite group commute? A formal answer, Pr2(G) = {(x, y) [element of] G2 |xy = yx}| / |G|2 begs our next question. How many ordered pairs of elements of a finite group commute

Rewriteability in Finite Groups

What\u27s the probability that two elements in a finite group commute? A formal answer, Pr2(G) = {(x, y) [element of] G2 |xy = yx}| / |G|2 begs our next question. How many ordered pairs of elements of a finite group commute

The Proportion of Fixed-Point-Free Elements of a Transitive Permutation Group

In 1990 Hendrik W. Lenstra, Jr. asked the following question: if G is a transitive permutation group of degree n and A is the set of elements of G that move every letter, then can one find a lower bound (in terms of n) for f(G) = |A|/|G|? Shortly thereafter, Arjeh Cohen showed that 1/n is such a bound. Lenstra’s problem arose from his work on the number field sieve. A simple example of how f(G) arises in number theory is the following: if h is an irreducible polynomial over the integers, consider the proportion: |{primes ≤ x | h has no zeroes mod p}| / |{primes ≤ x}| As x → ∞, this ratio approaches f(G), where G is the Galois group of h considered as a permutation group on its roots. Our results in this paper include explicit calculations of f(G) for groups G in several families. We also obtain results useful for computing f(G) when G is a wreath product or a direct product of permutation groups. Using this we show that {f(G) | G is transitive} is dense in [0, 1]. The corresponding conclusion is true if we restrict G to primitive groups

Algebraic geometric codes over rings

The techniques of algebraic geometry have been widely and successfully applied to the study of linear codes over finite fields since the early 1980's. Recently, there has been an increased interest in the study of linear codes over finite rings. In this thesis, we combine these two approaches to coding theory by introducing and studying algebraic geometric codes over rings.We define algebraic geometric codes over any local Artinian ring A and compute their parameters. Under the additional hypothesis that A is a Gorenstein ring, we show that the class of codes we have defined is closed under duals. We show that the coordinatewise projection of an algebraic geometric code defined over A is an algebraic geometric code defined over the residue field of A. As an example of our construction, we show that the linear \doubz/4-code which Hammons, et al., project nonlinearly to obtain the Nordstrom-Robinson code is an algebraic geometric code. In the case where A is either \doubz/q or a Galois ring, we find an expression for the minimum Euclidean weights of (trace codes of) certain algebraic geometric codes over A in terms of an exponential sum.U of I OnlyETDs are only available to UIUC Users without author permissio

A Numerical Approach to Rewriteability in Finite Groups

In this paper we compute the probability that an n-tuple for a group G is S-rewritable for a given set S of permutations for several classes of groups