Free products with amalgamation of finite groups and finite outer automorphism groups of free groups (Q1126547)

A non-trivial free product with amalgamation \(E=A*_UB\) of two finite groups \(A,B\) is said to be a \((p,q)\)-amalgam if \(| A:U|=p\), \(| B:U|=q\). The author proves that if a \((p,q)\)-amalgam \(E_0=A_0*_{U_0}B_0\), where \(p,q\) are non-equal numbers and \(U_0\) is a maximal subgroup in both amalgamated subgroups \(A_0\) and \(B_0\), is a subgroup of finite index \(m\) in a group \(E\) then \(E\) is also a \((p,q)\)-amalgam, indeed, \(E=A*_UB\) where \(A_0\leq A\), \(B_0\leq B\), \(A_0\cap U=B_0\cap U=U_0\) and \(| A:A_0|=| B:B_0|=| U:U_0|=m\). As an application, the author presents a method how to recognize and construct maximal finite subgroups of the outer automorphism group of a free group.