a finite group g is said to be a pos-group if for each x in g the cardinality of the set {y in g | o(y) = o(x)} is a divisor of the order of g. in this paper we study the structure of pos-groups with some cyclic sylow subgroups.

a finite group g is said to be a pos-group if for each x in g the cardinality of the set {y in g | o(y) = o(x)} is a divisor of the order of g. in this paper we study the structure of pos-groups with some cyclic sylow subgroups.

A finite group G is said to be a POS-group if for each x in G the cardinality of the set {y in G | o(y) = o(x)} is a divisor of the order of G. In this paper we study the structure of POS-groups with some cyclic Sylow subgroups.

the idea of “forcing” has long been used in many research fields, such as colorings, orientations, geodetics and dominating sets in graph theory, as well as latin squares, block designs and steiner systems in combinatorics (see [1] and the references therein). recently, the forcing on perfect matchings has been attracting more researchers attention. a forcing set of m is a subset of m contained...

The idea of “forcing” has long been used in many research fields, such as colorings, orientations, geodetics and dominating sets in graph theory, as well as Latin squares, block designs and Steiner systems in combinatorics (see [1] and the references therein). Recently, the forcing on perfect matchings has been attracting more researchers attention. A forcing set of M is a subset of M contained...

Let G be a finite group and let x ∈ G. Define the order subset of G determined by x to be the set of all elements in G that have the same order as x. A group G is said to have perfect order subsets if the number of elements in each order subset of G is a divisor of |G|. In this article we prove a theorem for a class of nonabelian groups, which is analogous to Theorem 4 in [2]. We then prove tha...

A perfect code in a graph Γ = (V,E) is a subset C of V that is an independent set such that every vertex in V \ C is adjacent to exactly one vertex in C. A total perfect code in Γ is a subset C of V such that every vertex of V is adjacent to exactly one vertex in C. A perfect code in the Hamming graph H(n, q) agrees with a q-ary perfect 1-code of length n in the classical setting. A necessary a...

We show that for every rooted, finitely branching, pruned tree T of height ω there exists a family F which consists of order isomorphic to T subtrees of the dyadic tree C = {0, 1}<N with the following properties: (i) the family F is a Gδ subset of 2C ; (ii) every perfect subtree of C contains a member of F ; (iii) if K is an analytic subset of F , then for every perfect subtree S of C there exi...

Let P be a perfect subset of the real line, and let the «-element subsets of P be partitioned into finitely many classes, each open (or just Borel) in the natural topology on the collection of such subsets. Then P has a perfect subset whose «-element subsets he in at most (n — 1)! of the classes. Let C be the set of infinite sequences of zeros and ones, topologized as the product of countably m...