# Automorphism group

In mathematics, the automorphism group of an object is the group of all its automorphisms, where the operation on them is function composition. Intuitively, this is the group of its "inner symmetries".

Of particular interest to polytopes is the automorphism group of an abstract polytope. This generalizes the symmetry groups of realizations – in fact, symmetry groups of polytopes are always subgroups of their automorphism groups. The modern study of polytopes has revolved largely around figuring out the behavior of these groups.

The automorphism groups of abstract polytopes act on them in two important ways: on their elements, and on their flags. In the latter case, the action is free, which is to say that no automorphism other than the identity leaves a flag in place.

## Distinguished generators

The distinguished generators of automorphism group of an abstract polytope are a generating set of the automorphism group ${\displaystyle \left\langle\rho_0,\dots,\rho_{n-1}\right\rangle}$ such that:

• Each generator ${\displaystyle \rho_i}$ is an involution (${\displaystyle \rho_i\rho_i = 1}$).
• Two generators ${\displaystyle \rho_i}$ and ${\displaystyle \rho_j}$ commute (${\displaystyle \rho_i\rho_j=\rho_j\rho_i}$) if ${\displaystyle 0\leq i\leq j-2\leq n-3}$.
• For any ${\displaystyle J,K \subseteq \{0,\dots,n-1\}}$, ${\displaystyle \left\langle\rho_i\mid i\in J\right\rangle \cap \left\langle\rho_i\mid i\in K\right\rangle = \left\langle\rho_i\mid i\in J\cap K\right\rangle}$.

This last property is called the intersection property.