Some results from Dassow and Mitrana (Internat. J. Comput. Algebra (2000)), Griebach (Theoret. Comput. Sci. 7 (1978) 311) and Ibarra et al. (Theoret. Comput. Sci. 2 (1976) 271) are generalized for finite autómata over arbitrary groups. The closure properties of these autómata are poorer and the accepting power is smaller when abelian groups are considered. We prove that the addition of any abel...

We show that ω-categorical rings with NIP are nilpotent-by-finite. We prove that an ω-categorical group with NIP and fsg is nilpotent-by-finite. We also notice that an ω-categorical group with at least one strongly regular type is abelian. Moreover, we get that each ω-categorical, characteristically simple p-group with NIP has an infinite, definable abelian subgroup. Assuming additionally the e...

Let $G$ be a finite group. We construct the prime graph of $ G $,which is denoted by $ Gamma(G) $ as follows: the vertex set of thisgraph is the prime divisors of $ |G| $ and two distinct vertices $ p$ and $ q $ are joined by an edge if and only if $ G $ contains anelement of order $ pq $.In this paper, we determine finite groups $ G $ with $ Gamma(G) =Gamma(L_3(q)) $, $2 leq q < 100 $ and prov...

The triple factorization of a group $G$ has been studied recently showing that $G=ABA$ for some proper subgroups $A$ and $B$ of $G$, the definition of rank-two geometry and rank-two coset geometry which is closely related to the triple factorization was defined and calculated for abelian groups. In this paper we study two infinite classes of non-abelian finite groups $D_{2n}$ and $PSL(2,2^{n})$...

Matrix embedding (ME) is a well-known steganographic scheme that can improve the embedding efficiency of steganography. In ME, the sender and recipient agree on a matrix in advance, and the message will be embedded into the cover data according to the matrix. In this paper, we propose a general framework for ME based on covering sequence (CS) of finite abelian group. By the proposed approach, t...

The problem of enumeration of conjugacy classes of finite abelian subgroups of the mapping class group Mσ of a compact closed surface X of genus σ is considered. A complete method of enumeration is achieved for finite elementary abelian subgroups and steps are taken toward enumera-tion of finite abelian subgroups.

In this article, we formalize that every finite cyclic group is isomorphic to a direct product of finite cyclic groups which orders are relative prime. This theorem is closely related to the Chinese Remainder theorem ([18]) and is a useful lemma to prove the basis theorem for finite abelian groups and the fundamental theorem of finite abelian groups. Moreover, we formalize some facts about the ...

These are some notes I have written up for myself. I have compiled what I feel to be some of the most important facts concerning characters on finite abelian groups, especially from a number theoretic point of view. I decided to distribute these because I couldn’t find a single reference that covered everything that I thought was relevant to the material I will discuss on Wednesday. Of course, ...

The main goal in this section is to prove the Fundamental Theorem of Abelian Groups, which roughly speaking says that every finite, abelian group is isomorphic to a product of cyclic groups. In other words, if A is a finite, abelian group, then there are positive integers, n1, . . . , nr, so that A ∼= Z/n1Z× · · · × Z/nrZ. It’s possible to have Z/n1Z × · · · × Z/nrZ ∼= Z/m1Z × · · · × Z/msZ whe...