In this paper we analyze possible actions of an automorphism of order six on a \(2\)-\((70, 24, 8)\) design, and give a complete classification for the action of the cyclic group of order six \(G= ...\langle \rho \rangle \cong Z_6 \cong Z_2 \times Z_3\), where \(\rho^3\) fixes exactly \(14\) points (blocks) and \(\rho^2\) fixes \(4\) points (blocks). Up to isomorphism there are \(3718\) such designs. This result significantly increases the number of
previously known \(2\)-\((70,24,8)\) designs.
The pairs (D,G), where D is a non-trivial 2-(k2,k,λ) design with λ|k and G is a flag-transitive automorphism group of D are classified except for k a power of a prime and G⩽AΓL1(k2).
Let $L$ be an algebra over a field $F$ with the binary operations $+$ and $,$. Then $L$ is called a left Leibniz algebra if it satisfies the left Leibniz identity: $a,b,c=a,b,c-b,a,c$ for all ...elements $a,b,c\in L$. A linear transformation $f$ of $L$ is called an endomorphism of $L$, if $f(a,b)=f(a),f(b)$ for all elements $a,b\in L$. A bijective endomorphism of $L$ is called an automorphism of $L$. It is easy to show that the set of all automorphisms of the Leibniz algebra is a group with respect to the operation of multiplication of automorphisms. The description of the structure of the automorphism groups of Leibniz algebras is one of the natural and important problems of the general Leibniz algebra theory. The main goal of this article is to describe the structure of the automorphism group of a certain type of nilpotent three-dimensional Leibniz algebras.
The symmetric 2-(v,k,λ) designs with k>λ(λ−3)/2 admitting a flag-transitive point-imprimitive automorphism group are completely classified: they are the known 2-designs with parameters ...(16,6,2),(45,12,3),(15,8,4) or (96,20,4).
A recent classification of flag-transitive 2-designs with parameters (v,k,λ) whose replication number r is coprime to λ gives rise to eight possible infinite families of 2-designs, some of which are ...with new parameters. In this note, we give explicit constructions for two of these families of 2-designs, and show that for a given positive integer q=32n+1⩾27, there exist 2-designs with parameters (q3+1,qi,qi−1), for i=1,2, admitting the Ree group 2G2(q) as their automorphism groups.
In this paper, all non-symmetric 2-(v,k,λ) designs with λ≥(r,λ)2 are classified which admit a flag-transitive group of automorphisms with an alternating socle. Up to isomorphism, our classification ...shows exactly 20 such designs, and, for each of these designs, we explicitly determine the corresponding flag-transitive automorphism groups.
For any positive integers n,s,t,l such that n≥10, s,t≥2, l≥1 and n≥s+t+l, a new infinite family of regular 3-hypertopes with type (2s,2t,2l) and automorphism group of order 2n is constructed.
