A note on barely transitive permutation groups satisfying
A survey on packing and covering problems in the Hamming permutation space.
Affine Weyl groups as infinite permutations.
Cayley's Theorem
The article formalizes the Cayley's theorem saying that every group G is isomorphic to a subgroup of the symmetric group on G.
Classification of Sn-normal semigroups.
Collineation group as a subgroup of the symmetric group
Let ψ be the projectivization (i.e., the set of one-dimensional vector subspaces) of a vector space of dimension ≥ 3 over a field. Let H be a closed (in the pointwise convergence topology) subgroup of the permutation group of the set ψ. Suppose that H contains the projective group and an arbitrary self-bijection of ψ transforming a triple of collinear points to a non-collinear triple. It is well known from [Kantor W.M., McDonough T.P., On the maximality of PSL(d+1,q), d ≥ 2, J. London Math. Soc.,...
Constructing a maximal cofinitary group.
Constructing matrix geometric means.
Coset representatives for permutation groups
Crossed product of cyclic groups
All crossed products of two cyclic groups are explicitly described using generators and relations. A necessary and sufficient condition for an extension of a group by a group to be a cyclic group is given.
Dihedral -tilings of the sphere by equilateral and scalene triangles. II.
Dihedral f-tilings of the sphere by equilateral and scalene triangles. III.
Dihedral f-tilings of the sphere by equilateral and scalene triangles-I
Every reasonably sized matrix group is a subgroup of S ∞
Every reasonably sized matrix group has an injective homomorphism into the group of all bijections of the natural numbers. However, not every reasonably sized simple group has an injective homomorphism into .
Geometric and combinatorial structure of a class of spherical folding tessellations – I
A classification of dihedral folding tessellations of the sphere whose prototiles are a kite and an equilateral or isosceles triangle was obtained in recent four papers by Avelino and Santos (2012, 2013, 2014 and 2015). In this paper we extend this classification, presenting all dihedral folding tessellations of the sphere by kites and scalene triangles in which the shorter side of the kite is equal to the longest side of the triangle. Within two possible cases of adjacency, only one will be addressed....
Highly transitive subgroups of the symmetric group on the natural numbers
Highly transitive subgroups of the symmetric group on the natural numbers are studied using combinatorics and the Baire category method. In particular, elementary combinatorial arguments are used to prove that given any nonidentity permutation α on ℕ there is another permutation β on ℕ such that the subgroup generated by α and β is highly transitive. The Baire category method is used to prove that for certain types of permutation α there are many such possibilities for β. As a simple corollary,...
Invariance groups of finite functions and orbit equivalence of permutation groups
Which subgroups of the symmetric group Sn arise as invariance groups of n-variable functions defined on a k-element domain? It appears that the higher the difference n-k, the more difficult it is to answer this question. For k ≤ n, the answer is easy: all subgroups of Sn are invariance groups. We give a complete answer in the cases k = n-1 and k = n-2, and we also give a partial answer in the general case: we describe invariance groups when n is much larger than n-k. The proof utilizes Galois connections...
Inversions of permutations in symmetric, alternating, and dihedral groups.
Maximal clones and maximal permutation groups
A fundamental result in universal algebra is the theorem of Rosenberg describing the maximal subclones in the clone of all operations over a finite set. In group theory, the maximal subgroups of the symmetric groups are classified by the O'Nan-Scott Theorem. We shall explore the similarities and differences between these two analogous major results. In addition, we show that a primitive permutation group of diagonal type can be maximal in the symmetric group only if its socle is the direct product...
Modular permutations on