# 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 $D$.