In this paper, by using the notion of fuzzy subsets, the concept of F-permutation is introduced. Then by applying this notion the concepts of presentation of an F-polygroup, graph of an F-permutation and F-permutation matrices are investigated.

A simplicial group is a object in the category of groups. very nice application which polygroup given this paper. Using polygroups instead groups, we already had good results from well known properties due to Loday. Loday proved that crossed module, cat1-group, categories and whose Moore complex length one are equivalent. Loday’s idea present functor groups polygroups. We show there exist cat1-...

Let Fn be the Farey sequence of order n. For S?Fn we let Q(S)={x/y:x,y?S,x?y and y?0}. We show that if Q(S)?Fn, then |S|?n+1. Moreover, prove in any following cases: (1) Q(S)=Fn; (2) Q(S)?Fn |S|=n+1, must have S={0,1,12,…,1n} or S={0,1,1n,…,n?1n} except for n=4, where an additional set {0,1,12,13,23} second case. Our results are based on Graham's GCD conjectures, which been proved by Balasubram...

Let Out(Fn) denote the outer automorphism group of the free group Fn with n > 3. We prove that for any finite index subgroup Γ < Out(Fn), the group Aut(Γ) is isomorphic to the normalizer of Γ in Out(Fn). We prove that Γ is co-Hopfian : every injective homomorphism Γ → Γ is surjective. Finally, we prove that the abstract commensurator Comm(Out(Fn)) is isomorphic to Out(Fn).

Let FN be the distribution function of a finite real population size N. Fn empirical sample n drawn from without replacement. TFN any product moments or cumulants FN, let TFn denote version, and Tn,NFN expected value with respect to FN. We prove following remarkable inversion principle that TN,nFn is equal TFN. also obtain an explicit expression for all orders up six.