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.

Two strings u and v are said to be Abelian equivalent if u is a permutation of the characters of v. We introduce two new regularities on strings w.r.t. Abelian equivalence, called Abelian covers and Abelian runs, which are generalizations of covers and runs of strings, respectively. We show how to determine in O(n) time whether or not a given string w of length n has an Abelian cover. Also, we ...

We start with a discussion of CM abelian varieties in characteristic zero, and in positive characteristic. An abelian variety over a finite field is a CM abelian variety, as Tate proved. Can it be CM lifted to characteristic zero? Here are other questions. Does there exist an abelian variety, say over Qa, or over Fp, of dimension g > 3 not isogenous with the Jacobian of an algebraic curve? Can ...

In this paper we give an elementary argument about the atoms and coatoms of the latticeof all subgroups of a group. It is proved that an abelian group of finite exponent is strongly coatomic.