The maximum order of finite groups of outer automorphisms of free groups (Q1329039)

Let \(F_ r\) be the free group of rank \(r>1\) and \(\text{Out }F_ r= \Aut F_ r/ \text{Inn } F_ r\), the outer automorphism group (automorphisms modulo inner automorphisms). We have the following Theorem: The maximum order of a finite subgroup \(G\) of \(\text{Out } F_ r\) is 12, for \(r=2\), and \(2^ r r!\), for \(r>2\). Furthermore the finite subgroup of \(\text{Out } F_ r\) realizing the maximum order is unique up to conjugacy, for \(r>3\).