Minimal odd order automorphism groups

We show that 3^7 is the smallest order of a non-trivial odd order group which occurs as the full automorphism group of a finite group.Comment: 11 pages, no figures. Manuscript accepted for publicatio

Commuting probability for subrings and quotient rings

We prove that the commuting probability of a finite ring is no larger than the commuting probabilities of its subrings and quotients, and characterize when equality occurs in such a comparison

Boolean rings are definitely commutative!

A ring {R, +, .} is called Boolean if r2 = r for all r ∈ R. We present four proofs that a Boolean ring is commutative

Counting commutativities in finite algebraic systems

We examine the total possible number of commutativities in a finite algebraic system, concentrating on groups, but also examining rings and semigroups. Numerical restrictions are found and bounds for the total number of commutativities in subgroups and factor groups are derived. Finally, a curious connection with group representations is explored

Converse Lagrange theorem orders and supersolvable orders

For finite groups, we investigate both converse Lagrange theorem (CLT) orders and supersolvable (SS) orders, and see that the latter form a proper subset of the former. We focus on the difference between these two sets of orders, reformulate the work of earlier authors algorithmically, and construct a computer program to enumerate such NSS-CLT orders. We establish several results relating to NSS and CLT orders and, working from our computer-generated data, propose a pair of conjectures and obtain a complete characterization of the most common form of NSS-CLT order

Noncommutative anticommutative rings

Abstract included in tex

Noncommutative anticommutative rings

Abstract included in tex

Small rings without ideal centres

We show that the smallest indecomposable non-unital ring in which the centre is not an ideal has order 3