On the Direct and Skew Sums of Γ1−Non Deranged Permutations

Abstract

Direct and skew sum operations are invaluable techniques for linking permutations while retaining their original structure in the resulting concatenation. In this work we apply the direct and skew sum operations on the elements of the Γ1−non deranged permutation group (GΓ1p),and present relations and schemes on the structures and fixed points ofthe permutations obtained from these operations. Furthermore, if π is thedirect sum of these Γ1− non deranged permutations, then the collectionof permutations in the form of π is an abelian group under composition,denoted as Gm⊕p. We present an expression relating the direct and skewsum operations, and we establish an isomorphism between GΓ1p × GΓ1p and Gm⊕p.