Let $${\mathcal {D}}$$ be a nontrivial 3-(v, k, 1) design admitting block-transitive group G of automorphisms. A recent work Gan and the second author asserts that is either affine or almost simple. In this paper, it proved if simple with socle an alternating group, then unique 3-(10, 4, design, $$G=\textrm{PGL}(2,9)$$ , $$\textrm{M}_{10}$$ $$\textrm{Aut}(\textrm{A}_6 )=\textrm{S}_6:\textrm{Z}_...

Let $G$ be a finite group and $pi(G)$ be the set of all prime divisors of $|G|$. The prime graph of $G$ is a simple graph $Gamma(G)$ with vertex set $pi(G)$ and two distinct vertices $p$ and $q$ in $pi(G)$ are adjacent by an edge if an only if $G$ has an element of order $pq$. In this case, we write $psim q$. Let $|G= p_1^{alpha_1}cdot p_2^{alpha_2}cdots p_k^{alpha_k}$, where $p_1

Problem 6.6 in the Kourovka Notebook [9], posed by H. Wielandt, reads as follows. ' Let P, Q be permutation representations of a finite group G with the same character. Suppose P(G) is a primitive permutation group. Is Q{G) necessarily primitive? Equivalently: Let A, B be subgroups of a finite group G such that for each class C of conjugate elements of G their intersection with C has the same c...

Many common finite p-groups admit automorphisms of order coprime to p, and when p is odd, it is reasonably difficult to find finite p-groups whose automorphism group is a p-group. Yet the goal of this paper is to prove that the automorphism group of a finite p-group is almost always a p-group. The asymptotics in our theorem involve fixing any two of the following parameters and letting the thir...

A longstanding open problem in the theory of von Neumann regular rings is the question of whether every directly finite simple regular ring must be unit-regular. Recent work on this problem has been done by P. Menal, K.C . O'Meara, and the authors. To clarify some aspects of these new developments, we introduce and study the notion of almost isomorphism between finitely generated projective mod...

One of the basic problems for Lie groups is to describe the image of the exponential map. A Lie group H is called exponential resp. weakly exponential if its exponential map is surjective resp. has dense image in H . In this paper we classify all exponential groups which are identity components of almost simple real algebraic groups. The weakly exponential groups of this type were determined in...