Let X and Y be infinite graphs, such that the automorphism group of X is nonamenable, and the automorphism group of Y has an infinite orbit. We prove that there is no automorphism-invariant measure on the set of spanning trees in the direct product X × Y . This implies that the minimal spanning forest corresponding to i.i.d. edge-weights in such a product, has infinitely many connected componen...

This survey on the automorphism groups of finite p-groups focuses on three major topics: explicit computations for familiar finite p-groups, such as the extraspecial p-groups and Sylow p-subgroups of Chevalley groups; constructing p-groups with specified automorphism groups; and the discovery of finite p-groups whose automorphism groups are or are not p-groups themselves. The material is presen...

a $p$-group $g$ is $p$-central if $g^{p}le z(g)$‎, ‎and $g$ is‎ ‎$p^{2}$-abelian if $(xy)^{p^{2}}=x^{p^{2}}y^{p^{2}}$ for all $x,yin‎ ‎g$‎. ‎we prove that for $g$ a finite $p^{2}$-abelian $p$-central‎ ‎$p$-group‎, ‎excluding certain cases‎, ‎the order of $g$ divides the‎ ‎order of $text{aut}(g)$‎.

let $g$ be a $p$-group of order $p^n$ and $phi$=$phi(g)$ be the frattini subgroup of $g$. it is shown that the nilpotency class of $autf(g)$, the group of all automorphisms of $g$ centralizing $g/ fr(g)$, takes the maximum value $n-2$ if and only if $g$ is of maximal class. we also determine the nilpotency class of $autf(g)$ when $g$ is a finite abelian $p$-group.

let r and s be reduced rings with identities whose idempotents are central, and let m be an(r, s)-bimodule such that annr (m)=0. in this paper, we determine first the structure of automorphisms ofthe triangular ring, =sr mt0, and then, for all automorphisms α ,β of t we determine the structureof (α,β)-derivations of t.

There are 172,248 Steiner triple systems of order 19 having a nontrivial automorphism group. Computational methods suitable for generating these designs are developed. The use of tactical configurations in conjunction with orderly algorithms underlies practical techniques for the generation of the designs, and the subexponential time isomorphism technique for triple systems is improved in pract...

We investigate the extent to which Slaman and Woodin’s framework for the analysis of the automorphism group of the structure of the Turing degrees can be transferred to analyze the automorphism group of the structure of the enumeration degrees.

We show that certain finite groups do not arise as the automorphism group of the square of a finite algebraic structure, nor as the automorphism group of a finite, 2-generated, free, algebraic structure.

Let R be a commutative ring and S a multiplicatively closed subset of R having no zero divisors. The pair (.R, S) is said to be stable if the ring of fractions of R, S R, defined by S is a ring for which all finitely generated projective modules are free. For a stable pair (R, S) assume 2 is a unit in R and V is a free R-module of dimension > 3. This paper examines the action of a group automor...

The paper studies line-transitive, point-imprimitive automorphism groups G of finite linear spaces. In particular, it explores inequalities involving two integer parameters x; y introduced by Delandtsheer and Doyen associated with a given G-invariant partition C of the point set. There is special interest in the case where C is G-normal, that is, C is the set of orbits of a normal subgroup of G...