In this paper, we will determine the full automorphism groups of rose window graphs that are not edge-transitive. As the full automorphism groups of edge-transitive rose window graphs have been determined, this will complete the problem of calculating the full automorphism group of rose window graphs. As a corollary, we determine which rose window graphs are vertex-transitive. Finally, we deter...

There are 50,024 Kirkman triple systems of order 21 admitting an automorphism of order 2. There are 13,280 Kirkman triple systems of order 21 admitting an automorphism of order 3. Together with the 192 known systems and some simple exchange operations, this leads to a collection of 63,745 nonisomorphic Kirkman triple systems of order 21. This includes all KTS(21)s having a nontrivial automorphi...

The problem of finding minimal vertex number of graphs with a given automorphism group is addressed in this article for the case of cyclic groups. This problem was considered earlier by other authors. We give a construction of an undirected graph having 4n+ 6 vertices and automorphism group cyclic of order 4n, n ≥ 1. As a special case we get graphs with 2k + 6 vertices and cyclic automorphism g...

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...

The Fischer group Fi22 acts as a rank 3 group of automorphisms of a symmetric 2-(14080,1444,148) design. This design does not have a doubly transitive automorphism group, since there is a partial linear space with lines of size 4 defined combinatorially from the design and preserved by its automorphism group. We investigate this geometry and determine the structure of various subspaces of it. I...

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.