Factorization number and subgroup commutativity degree via spectral invariants

• Published in: Computational & applied mathematics Vol. 42; no. 3
• Main Authors: Muhie, Seid Kassaw, Otera, Daniele Ettore, Russo, Francesco G.
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 01.04.2023; Springer Nature B.V
• Subjects: Apexes; Applications of Mathematics; Applied physics; Combinatorial analysis; Commutativity; Computational mathematics; Computational Mathematics and Numerical Analysis; Edge joints; Factorization; Graph theory; Mathematical Applications in Computer Science; Mathematical Applications in the Physical Sciences; Mathematics; Mathematics and Statistics; Subgroups; Subgroup commutativity degree; 20K27; Non-permutability graph of subgroups; 05C07; Factorization number; Laplacian matrix; Primary 20D60; 05C25; Spectrum; Secondary 05C15
• DOI: 10.1007/s40314-023-02270-5
• ISSN: 2238-3603

The factorization number F 2 ( G ) of a finite group G is the number of all possible factorizations of G = H K as a product of its subgroups H and K , while the subgroup commutativity degree sd ( G ) of G is the probability of finding two commuting subgroups in G at random. It is known that sd ( G ) can be expressed in terms of F 2 ( G ) . Denoting by L ( G ) the subgroups lattice of G , the non–permutability graph of subgroups Γ L ( G ) of G is the graph with vertices in L ( G ) \ C L ( G ) ( L ( G ) ) , where C L ( G ) ( L ( G ) ) is the smallest sublattice of L ( G ) containing all permutable subgroups of G , and edges obtained by joining two vertices X ,  Y such that X Y ≠ Y X . The spectral properties of Γ L ( G ) have been recently investigated in connection with F 2 ( G ) and sd ( G ) . Here we show a new combinatorial formula, which allows us to express F 2 ( G ) , and so sd ( G ) , in terms of adjacency and Laplacian matrices of Γ L ( G ) .