let be a set and let be the set of subsets of . the pair in which is a collection of elements of (blocks) is called a design if every element of appears in , times. is called a symmetric design if . in a symmetric design, each element of appears times in blocks of . a mapping between two designs and is an isomorphism if is a one-to-one correspondence and . every isomorphism of a design, , to it...

In this paper, we consider the projective special linear group $PSL_2(59)$ and construct some 1-designs by applying the Key-Moori method on $PSL_2(59)$. Moreover, we obtain parameters of these designs and their automorphism groups. It is shown that $PSL_2(59)$ and $PSL_2(59):2$ appear as the automorphism group of the constructed designs.

Let $G$ be a finite group. The automorphism $sigma$ of a group $G$ is said to be an absolute central automorphism, if for all $xin G$, $x^{-1}x^{sigma}in L(G)$, where $L(G)$ be the absolute centre of $G$. In this paper, we study  some properties of absolute central automorphisms of a given finite $p$-group.

We present 15 new partial difference sets over 4 non-abelian groups of order 100 and 2 new strongly regular graphs with intransitive automorphism groups. The strongly regular graphs and corresponding partial difference sets have the following parameters: (100,22,0,6), (100,36,14,12), (100,45,20,20), (100,44,18,20). The existence of strongly regular graphs with the latter set of parameters was a...

Let G be a finite simple graph of order n, maximum degree ∆, and minimum degree δ. A compact regularization of G is a ∆-regular graph H of which G is an induced subgraph: H is symmetric if every automorphism of G can be extended to an automorphism of H. The index |H : G| of a regularization H of G is the ratio |V (H)|/|V (G)|. Let mcr(G) denote the index of a minimum compact regularization of G...

This article describes the use of permutation representations to obtain a partial solution to the Extensible Automorphisms Problem. It introduces a notion of permutation-extensibility and proves the equivalence of this notion with the notion of being subgroup-conjugating. 1. The problem we want to solve 1.1. Three big problems. A quick recall of three definitions: Definition. (1) An automorphis...

If G is a finite subgroup of the automorphism group of a projective curve X and D is a divisor on X stabilized by G, then we compute a simplified formula for the trace of the natural representation of G on the Riemann-Roch space L(D), under the assumption that L(D) is “rational”, D is nonspecial, and the characteristic is “good”. We discuss the partial formulas that result if L(D) is not rational.

We prove induced Ramsey theorems in which the monochromatic induced subgraph satisfies that all members of a prescribed set of its partial isomorphisms extend to automorphisms of the colored graph (without requirement of preservation of colors). We consider vertex and edge colorings, and extensions of partial isomorphisms in the set of all partial isomorphisms between singletons as considered b...

A comparison technique for random walks on finite graphs is introduced, using the well-known interlacing method. It yields improved return probability bounds. A key feature is the incorporation of parts of the spectrum of the transition matrix other than just the principal eigenvalue. As an application, an upper bound of the expected return probability of a random walk with symmetric transition...