The concept of a coset-preserving skew-morphism is a generalization of the widely studied t-balanced skew-morphisms of regular Cayley maps which are in turn generalizations of group automorphisms. In case of abelian groups, all skew-morphisms of regular Cayley maps are roots of coset-preserving skew-morphisms, and therefore, classification of cosetpreserving skew-morphisms of finite abelian gro...

We survey properties of the set of possible exponents of subsets of Zn (equivalently, exponents of primitive circulant digraphs on n vertices). Let En denote this exponent set. We point out that En contains the positive integers up to √ n, the ‘large’ exponents ⌊ 3 ⌋+1, ⌊ 2 ⌋, n−1, and for even n ≥ 4, the additional value n 2 −1. It is easy to see that no exponent in [n 2 +1, n−2] is possible, ...

In this paper we classify finite groups whose nonnormal subgroups are of order p or pq, where p, q are primes. As a by-product, we also classify the finite groups in which all nonnormal subgroups are cyclic.

We classify the (finite and infinite) virtually cyclic subgroups of the pure braid groups PnpRP2q of the projective plane. The maximal finite subgroups of PnpRP2q are isomorphic to the quaternion group of order 8 if n 3, and to Z4 if n ¥ 4. Further, for all n ¥ 3, up to isomorphism, the following groups are the infinite virtually cyclic subgroups of PnpRP2q: Z, Z2 Z and the amalgamated product ...

We show that among all finite groups of any given order, the cyclic group of that order has the maximum number of edges in its power graph.

Surface codes describe quantum memory stored as a global property of interacting spins on a surface. The state space is fixed by a complete set of quasi-local stabilizer operators and the code dimension depends on the first homology group of the surface complex. These code states can be actively stabilized by measurements or, alternatively, can be prepared by cooling to the ground subspace of a...

in this paper we provide explicit formulas for the number of elements/subgroups/cyclic subgroups of a given order and for the total number of subgroups/cyclic subgroups in a finite hamiltonian group. the coverings with three proper subgroups and the principal series of such a group are also counted. finally, we give a complete description of the lattice of characteristic subgroups of a finite h...

We have been working on the formalization of groups. In [1], we encoded some theorems concerning the product of cyclic groups. In this article, we present the generalized formalization of [1]. First, we show that every finite commutative group which order is composite number is isomorphic to a direct product of finite commutative groups which orders are relatively prime. Next, we describe finit...

We have been working on the formalization of groups. In [1], we encoded some theorems concerning the product of cyclic groups. In this article, we present the generalized formalization of [1]. First, we show that every finite commutative group which order is composite number is isomorphic to a direct product of finite commutative groups which orders are relatively prime. Next, we describe finit...