Automorphism Group and Correlation Group

These function are currently only available for incidence geometries.

An automorphism \(\alpha\) of an incidence geometry \(\Gamma(X,\sim,t,I)\) is an automorphism of the incidence graph of \(\Gamma\) such that for all \(x\in X\), \(t(\alpha(x)) = t(x)\). In other words, an automorphism cannot change the type of an element. The automorphism group of \(\Gamma\), denoted \(Aut(\Gamma)\), is the group of all automorphisms of \(\Gamma\).

A correlation \(\alpha\) of an incidence geometry \(\Gamma(X,\sim,t,I)\) is an automorphism of the incidence graph of \(\Gamma\) such that for all \(x, y\in X\), \(t(x) = t(y) \Rightarrow t(\alpha(x)) = t(\alpha(y))\). The correlation group of \(\Gamma\), denoted \(Cor(\Gamma)\), is the group of all correlations of \(\Gamma\).

It is obvious that \(Aut(\Gamma)\) is a subgroup of \(Cor(\Gamma)\).

For an incidence geometry \(\Gamma\), we can compute \(Aut(\Gamma)\) and \(Cor(\Gamma)\) using the commands described below.

AutomorphismGroup(D): IncGeom -> GrpPerm

Given an incidence geometry \(D\), return the group of type–preserving automorphisms of \(D\) as a permutation group of type GrpPerm acting on the set of elements of \(D\).

CorrelationGroup(D): IncGeom -> GrpPerm

Given an incidence geometry \(D\), return the group of automorphisms of \(D\) as a permutation group of type GrpPerm acting of the set on elements of \(D\).