This paper deals with the number of subgroups of a given exponent in a finite abelian group. Explicit formulas are obtained in the case of rank two and rank three abelian groups. An asymptotic formula is also presented.

It has been shown that the symmetry-protected topological (SPT) phases with finite Abelian symmetries can be described by Chern-Simons field theory. We propose a topological response theory to uniquely identify the SPT orders, which allows us to obtain a systematic scheme to classify bosonic SPT phases with any finite Abelian symmetry group. We point out that even for finite Abelian symmetry, t...

Baer characterized capable finite abelian groups (a group is capable if it is isomorphic to the quotient of some group by its center) by a condition on the size of the factors in the invariant factor decomposition (the group must be noncyclic and the top two invariant factors must be equal). We provide a different characterization, given in terms of a condition on the lattice of subgroups. Name...

Let R be a noetherian ring, and G(R) the Grothendieck group of finitely generated modules over R. For a finite abelian group n, we describe G(Rn) as the direct sum of groups G(R’). Each R’ is the form RI<,,, I/n], where n is a positive integer and Cn a primitive nth root of unity. As an application, we describe the structure of the Grothendieck group of pairs (H. u), where His an abelian group ...

Let G be a properly face 2-coloured (say black and white) piecewise-linear triangulation of the sphere with vertex set V . Consider the abelian group AW generated by the set V , with relations r+c+s = 0 for all white triangles with vertices r, c and s. The group AB can be defined similarly, using black triangles. These groups are related in the following manner AW ∼= AB ∼= Z⊕ Z⊕ C where C is a ...

This article gives the proof of results announced in [Bla07a] and some description of automorphisms of rational surfaces. Given a finite abelian subgroup of the Cremona group of the plane, we give a way to decide whether this one is birationally conjugate to a group of automorphisms of a minimal surface. In particular, we prove that a finite cyclic group of birational transformations of the pla...

This paper, deals with some equivalence relations in fuzzy subgroups. Further the probability of commuting two fuzzy subgroups of some finite abelian groups is defined.

Let FG be a group algebra of a finite non-abelian pgroup G and F a field of characteristic p. In this paper we give all minimal non-abelian p-groups and minimal non-metacyclic p-groups whose group algebras FG possess a filtered multiplicative F -basis.