Kneser’s theorem in -finite abelian groups

Lecture # 14: Abstract Algebra Fundamental Theorem of Finite Abelian Groups

Hindman’s Theorem in Abelian Groups

Hindman’s theorem tells us that for every finite colouring/partition of the natural numbers, there is an infinite subset X for which every finite subset X0 Ď X has the same colour for ř X0. Certain generalisations of this theorem to abelian groups are known to fail in the uncountable case but an example exists for boolean groups when the desired homogeneous set is finite. In this paper, I will ...

Decomposing finite Abelian groups

This paper describes a quantum algorithm for efficiently decomposing finite Abelian groups. Such a decomposition is needed in order to apply the Abelian hidden subgroup algorithm. Such a decomposition (assuming the Generalized Riemann Hypothesis) also leads to an efficient algorithm for computing class numbers (known to be at least as difficult as factoring).

Factoring finite abelian groups

It is shown that if in the factorization of a finite cyclic group each factor has either prime power order or order equal to the product of two primes then one factor must be periodic. This is shown to be the best possible result of this type.  2004 Elsevier Inc. All rights reserved.

On non-normal non-abelian subgroups of finite groups

‎In this paper we prove that a finite group $G$ having at most three‎ ‎conjugacy classes of non-normal non-abelian proper subgroups is‎ ‎always solvable except for $Gcong{rm{A_5}}$‎, ‎which extends Theorem 3.3‎ ‎in [Some sufficient conditions on the number of‎ ‎non-abelian subgroups of a finite group to be solvable‎, ‎Acta Math‎. ‎Sinica (English Series) 27 (2011) 891--896.]‎. ‎Moreover‎, ‎we s...