Consider a finite group G and subgroups H,K of G. We say that H and K permute if HK = KH and call H a permutable subgroup if H permutes with every subgroup of G. A group G is called quasi-Dedekind if all subgroups of G are permutable. We can define, for every finite group G, an arithmetic quantity that measures the probability that two subgroups (chosen uniformly at random with replacement) per...

If H is a subgroup of the symmetric group $4, then a 3-groupoid (S, f ) is called H-permutable if f ( x ~ ( 1 ) , X a ( 2 ) , X ~ r ( 3 ) ) = X~(4) ~ f ( x l , x2 ,x3 ) = x4 for every cr C H. Some classes of generalized idempotent H-permutable 3-groupoids are equivalent to Steiner, Mendelsohn and other quadruple systems. We prove that the variety of generalized idempotent H-permutable 3-groupoi...

let $h$, $l$ and $x$ be subgroups of a finite group$g$. then $h$ is said to be $x$-permutable with $l$ if for some$xin x$ we have $al^{x}=l^{x}a$. we say that $h$ is emph{$x$-quasipermutable } (emph{$x_{s}$-quasipermutable}, respectively) in $g$ provided $g$ has a subgroup$b$ such that $g=n_{g}(h)b$ and $h$ $x$-permutes with $b$ and with all subgroups (with all sylowsubgroups, respectively) $v$...

Let ? be a finite, undirected, connected, simple graph. We say that matching M is permutable m -matching if contains edges and the subgroup of Aut ( ) fixes setwise allows to permuted in any fashion. A 2-transitive stabilizer can map ordered pair distinct other . provide constructions graphs with matching; we show that, an arc-transitive graph for ? 4 , then degree at least ; and, when sufficie...

Abstract A group is called quasihamiltonian if all its subgroups are permutable, and we say that a subgroup Q of G permutably embedded in $\langle Q,g\rangle $ for each element g . It proved here contains normal such $G/Q$ ?ernikov, then has finite index; moreover, periodic, it ?ernikov N $G/N$ quasihamiltonian. This result should be compared with theorems Schlette stating over centre, abelian-...

A well known theorem of Murskiı̆’s asserts that almost every finite, nonunary algebra is idemprimal. We derive an analagous result under the assumption that all basic operations are idempotent. If the algebra contains a basic l-ary idempotent operation with l > 2 then the algebra is idemprimal with probability 1. However, for an algebra with a single basic binary operation, the probability of id...

One of the important classes of varieties identified in tame congruence theory is the class of varieties which are n-permutable for some n. In this paper we prove two results: (1) For every n > 1 there is a polynomial-time algorithm which, given a finite idempotent algebra A in a finite language, determines whether the variety generated by A is n-permutable; (2) A variety is n-permutable for so...

A diagrammatic statement is developed for the generalized semidistributive law in case of single algebras assuming that their congruences are permutable. Without permutable congruences, a diagrammatic statement is developed for the ∧-semidistributive law.