Permutable prime

Abstract: Let ^ be a family of subsets of a group G with the following three properties: (i) G is the union of the subsets in £C. (ii) Every two subsets in § commute. (iii) Every F e £C has cardinal (strictly) less than a fixed (finite or infinite) cardinal rt. We ask what can be said about the group G when the bound rt is given. Thus e.g. if 11 = 2, that is if all F e % consist of at most a single element, G clearly must be abelian; and conversely every abelian group can be covered by permutable subsets consisting of a single element each. The main result deals with the case that rt is finite. If H denotes the union of all finite classes of conjugate elements of G, then H is easily seen to be a subgroup of G. This can be shown to have index < u in G if 11 is finite; moreover the finite classes of conjugates of G, that is the classes of which H consists, are then boundedly finite. Using known results on groups in which all classes of conjugate elements are finite, one can then show that the derived group H' of H is finite. The converse is also true, and these facts can be combined in the following simple criterion: The group G can be covered by permutable, boundedly finite subsets if, and only if, G has a subgroup of finite index with finite derived group. This generalizes an unpublished result of F. I. Mautner, namely that if G possesses a finite subgroup K whose double cosets in G permute, then H (defined as above) has finite index in G. In fact we can show by elementary means that G = HK. For infinite values of n our results are much less complete. If rt = ^0» then G is covered by permutable finite, but not necessarily boundedly finite, subsets. All finitely generated groups and all countable locally finite groups are of this kind; but Paul M. Colin has recently shown that there are countable groups which are not of this kind, e.g. free groups with fc$0 free generators. The only general result that has so far been obtained is this: / / the order of G is strictly greater than rt, then G has a subgroup C whose order is also strictly greater than rt and whose centre is not trivial.

Abstract: Let us call an equational class (variety) K of algebras permutable if and only if every pair of congruences on each K-algebra is permutable. Similarly, we will call K modular (distributive) if the congruence lattice of each K-algebra is modular (distributive).

Abstract: A cyclically permutable code is a binary code whose codewords are cyclically distinct and have full cyclic order. An important class of these codes are the constant weight cyclically permutable codes. In a code of this class all codewords have the same weight w. These codes have many applications, in. Eluding in optical code-division multiple-access communication systems and in constructing protocol-sequence sets for the M-active-out-of-T users collision channel without feedback. In this paper we construct optimal constant weight cyclically permutable codes with length n, weight w, and a minimum Hamming distance 2w-2. Some of these codes coincide with the well-known design called a difference family. Some of the constructions use combinatorial structures with other applications in coding. >