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The correspondence between perfect difference sets and transitive projective planes is well-known. We observe that all known dense (i.e., close to square-root size) Sidon subsets of abelian groups come from projective planes through a similar construction. We classify the Sidon sets arising in this manner from desarguesian planes and find essentially no new examples. There are many further examples arising from nondesarguesian planes.We conjecture that all dense Sidon sets arise from finite projective planes in this way. If true, this implies that all abelian groups of most orders do not have dense Sidon subsets. In particular if $\sigma_n$ denotes the size of the largest Sidon subset of $\mathbb{Z}/n\mathbb{Z}$, this implies $\liminf_{n \to \infty} \sigma_n / n^{1/2} < 1$.We also give a brief bestiary of somewhat smaller Sidon sets with a variety of algebraic origins, and for some of them provide an overarching pattern.

Without using the classification of finite simple groups (CFSG), we show that the probability that two random elements of Sn$$ {S}_n $$ generate a primitive group smaller than An$$ {A}_n $$ is at most exp(−c(nlogn)1/2)$$ \exp \left(-c{\left(n\log n\right)}^{1/2}\right) $$. As a corollary we get Dixon's asymptotic expansion 1−1/n−1/n2−4/n3−23/n4−⋯$$ 1-1/n-1/{n}^2-4/{n}^3-23/{n}^4-\cdots $$for the probability that two random elements of Sn$$ {S}_n $$ (or An$$ {A}_n $$) generate a subgroup containing An$$ {A}_n $$. [ABSTRACT FROM AUTHOR]

A family of vectors in [k]n is said to be intersecting if any two of its elements agree on at least one coordinate. We prove, for fixed k ≥ 3, that the size of any intersecting subfamily of [k]n invariant under a transitive group of symmetries is o(kn), which is in stark contrast to the case of the Boolean hypercube (where k = 2). Our main contribution addresses limitations of existing technology: while there are now methods, first appearing in work of Ellis and the third author, for using spectral machinery to tackle problems in extremal set theory involving symmetry, this machinery relies crucially on the interplay between up-sets, biased product measures, and threshold behaviour in the Boolean hypercube, features that are notably absent in the problem considered here. To circumvent these barriers, introducing ideas that seem of independent interest, we develop a variant of the sharp threshold machinery that applies at the level of products of posets. [ABSTRACT FROM AUTHOR]

Let ${\mathbb{P}}(ord\pi = ord\pi ')$ be the probability that two independent, uniformly random permutations of [n] have the same order. Answering a question of Thibault Godin, we prove that ${\mathbb{P}}(ord\pi = ord\pi ') = {n^{ - 2 + o(1)}}$ and that ${\mathbb{P}}(ord\pi = ord\pi ') \ge {1 \over 2}{n^{ - 2}}lg*n$ for infinitely many n. (Here lg*n is the height of the tallest tower of twos that is less than or equal to n.) [ABSTRACT FROM AUTHOR]

It is well known that the proportion of pairs of elements of SL(n,q) which generate the group tends to 1 as qn → ∞. This was proved by Kantor and Lubotzky using the classification of finite simple groups. We give a proof of this theorem which does not depend on the classification. An essential step in our proof is an estimate for the average of (ord g)−1 when g ranges over GL (n,q), which may be of independent interest. We prove that this average is exp(−(2−o(1)) √n log n log q). [ABSTRACT FROM AUTHOR]

We prove an asymptotic for the number of additive triples of bijections (1,...,n) → ℤ/nℤ, that is, the number of pairs of bijections π1,π2:(1,...n) → ℤ/nℤ such that the pointwise sum π1 + π2 is also a bijection. This problem is equivalent to counting the number of orthomorphisms or complete mappings of ℤ/nℤ, to counting the number of arrangements of n mutually nonattacking semiqueens on an n x n toroidal chessboard, and to counting the number of transversals in a cyclic Latin square. The method of proof is a version of the Hardy-Littlewood circle method from analytic number theory, adapted to the group (ℤ /nℤ)n. [ABSTRACT FROM AUTHOR]

Let i(n, k) be the proportion of permutations π ∊ Sn having an invariant set of size k. In this note, we adapt arguments of the second author to prove that i(n, k) ≍k-δ(1 + log k) -3/2 uniformly for 1 ≤ k≤ n/2, where δ =1 - 1+log log 2/log 2. As an application, we show that the proportion of π ∊ Sn contained in a transitive subgroup not containing An is at least n-δ+o(1) if nis even. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to establish a method of obtaining closed-form solutions in isotropic hyperelasticity using the complementary energy, the Legendre transform of the strain energy function. Using the complementary energy, the stress becomes the independent variable and the strain the dependent variable. Some of the implications of this formulation of the equations are explored and illustrative examples of solutions for spherical and cylindrical inflation for several forms of the complementary energy are presented. [ABSTRACT FROM AUTHOR]

Here is a simplified proof that every sharply transitive subset of PGL2(K) is a coset of a subgroup, for every finite field K. [ABSTRACT FROM AUTHOR]

The commuting probability of a finite group is defined to be the probability that two randomly chosen group elements commute. Let P ⊂ (0,1] be the set of commuting probabilities of all finite groups. We prove that every point of P is nearly an Egyptian fraction of bounded complexity. As a corollary, we deduce two conjectures of Keith Joseph from 1977: all limit points of P are rational, and P is well ordered by >. We also prove analogous theorems for bilinear maps of abelian groups. [ABSTRACT FROM AUTHOR]

Folkman's theorem asserts that for each [ABSTRACT FROM AUTHOR]

We generalise an argument of Leader, Russell, and Walters to show that almost all sets of $d+ 2$ points on the $(d- 1)$-sphere ${S}^{d- 1} $ are not contained in a transitive set in some ${\mathbf{R} }^{n} $. [ABSTRACT FROM PUBLISHER]