Perfect numbers and finite groups
Periodic factorization of a finite Abelian 2-group.
Permutability of centre-by-finite groups
Let be a group and be an integer greater than or equal to . is said to be -permutable if every product of elements can be reordered at least in one way. We prove that, if has a centre of finite index , then is -permutable. More bounds are given on the least such that is -permutable.
Polynomial and Term Functions over Groups with Coprime Chief Factors.
Polynomials over the permutation group of three elements
Prime divisors of conjugacy class lengths in finite groups
We show that in a finite group which is -nilpotent for at most one prime dividing its order, there exists an element whose conjugacy class length is divisible by more than half of the primes dividing .
Prime graph components of finite almost simple groups
Primitive characters of subgroups of M-groups.
Probability that an element of a finite group has a square root
Let G be a finite group of even order. We give some bounds for the probability p(G) that a randomly chosen element in G has a square root. In particular, we prove that p(G) ≤ 1 - ⌊√|G|⌋/|G|. Moreover, we show that if the Sylow 2-subgroup of G is not a proper normal elementary abelian subgroup of G, then p(G) ≤ 1 - 1/√|G|. Both of these bounds are best possible upper bounds for p(G), depending only on the order of G.
Product decompositions of quasirandom groups and a Jordan type theorem
We first note that a result of Gowers on product-free sets in groups has an unexpected consequence: If is the minimal degree of a representation of the finite group , then for every subset of with we have . We use this to obtain improved versions of recent deep theorems of Helfgott and of Shalev concerning product decompositions of finite simple groups, with much simpler proofs. On the other hand, we prove a version of Jordan’s theorem which implies that if , then has a proper subgroup...
Properties of element orders in covers for and .