Abstract
Suppose that G is a group, X a subset of G and $$\pi $$? a set of natural numbers. The $$\pi $$?-product graph $$\mathcal {P}_{\pi }(G,X)$$P?(G,X) has X as its vertex set and distinct vertices are joined by an edge if the order of their product is in $$\pi $$?. If X is a set of involutions, then $$\mathcal {P}_{\pi }(G,X)$$P?(G,X) is called a $$\pi $$?-product involution graph. In this paper we study the connectivity and diameters of $$\mathcal {P}_{\pi }(G,X)$$P?(G,X) when G is a finite symmetric group and X is a G-conjugacy class of involutions.
Highlights
There is a cornucopia of combinatorial and geometric structures which are associated with groups. These range from graphs to posets and topological spaces such as simplicial complexes
Graphs and Combinatorics (2016) 32:1545–1570 associated order complex |Sp(G)| known as the Brown complex, after being studied by—among others—Brown in his paper [10]
Thévenez and Webb later showed that the complexes consisting of chains of normal series of p-subgroups, and chains of radical p-subgroups are G-homotopy equivalent to the Brown and Quillen complexes
Summary
There is a cornucopia of combinatorial and geometric structures which are associated with groups. There exists yi j ∈ H , where H := Sym(supp(ti , t j , xi , x j )), which is H -conjugate to ti t j and such that the connected components of the x-graphs Gytiitjj and Suppose that Gx has connected components Ci , C j and Ck of the given forms and define H := Sym(supp(ti , t j , tk, xi , x j , xk)).
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
