This paper investigates the automorphism group of a connected and undirected G-symmetric graph Γ where G is an almost simple group with socle T . First we prove that, for an arbitrary subgroup M of AutΓ containing G, either T is normal in M or T is a subgroup of the alternating group Ak of degree k = |Mα : Tα| − |NM (T ) : T |. Then we describe the structure of the full automorphism group of G-...

in this paper we introduce the concept of α-commutator which its definition is based on generalized conjugate classes. with this notion, α-nilpotent groups, α-solvable groups, nilpotency and solvability of groups related to the automorphism are defined. n(g) and s(g) are the set of all nilpotency classes and the set of all solvability classes for the group g with respect to different automorphi...

Abstract We estimate the number of graphical regular representations (GRRs) a given group with large enough order. As consequence, we show that almost all finite Cayley graphs have full automorphism groups ‘as small as possible’. This confirms conjecture Babai–Godsil–Imrich–Lovász on proportion GRRs, well Xu normal graphs, among group.

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...

Suppose that Φ is a reduced irreducible root system, R is an associative commutative ring with unity, G(Φ, R) is the corresponding adjoint Chevalley group, and E(Φ, R) is its elementary subgroup (see Section 5). There are a lot of results (see, e.g., [Wat80], [Pet82], [GMi83], [HO’M89], [Abe93], [Che00], [Bun07], and references therein*) asserting that, under some conditions, all automorphisms ...

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.

Let p be a prime, K a field of characteristic p , G a locally finite p-group, KG the group algebra, and V the group of the units of KG with augmentation 1. The anti-automorphism g 7→ g of G extends linearly to KG ; this extension leaves V setwise invariant, and its restriction to V followed by v 7→ v gives an automorphism of V . The elements of V fixed by this automorphism are called unitary; t...

In this paper we find a unique normal form for the symplectic matrix representation of the conjugacy class of a prime order element of the mapping-class group. We find a set of generators for the fundamental group of a surface with a conformal automorphism of prime order which reflects the action the automorphism in an optimal way. This is called an adapted homotopy basis and there is a corresp...