A class of finite group-amalgams

Abstract

Let A − 1 {A_{ - 1}} and A 1 {A_1} be finite groups such that A − 1 ∩ A 1 = A 0 {A_{ - 1}} \cap {A_1} = {A_0} is a common subgroup with [ A − 1 : A 0 ] = 4 [{A_{ - 1}}:{A_0}] = 4 and [ A 1 : A 0 ] = 2 [{A_1}:{A_0}] = 2 . We further assume that only the trivial subgroup of A 0 {A_0} is normal in both A − 1 {A_{ - 1}} and A 1 {A_1} . Let K be the intersection of all conjugates x A 0 x − 1 x{A_0}{x^{ - 1}} for x ∈ A − 1 x \in {A_{ - 1}} . Then if A 0 ≠ { 1 } {A_0} \ne \{ 1\} we have A − 1 / K ≅ D 4 , A 4 {A_{ - 1}}/K \cong {D_4},{A_4} , or S 4 {S_4} . We describe in detail all such amalgams ( A − 1 , A 1 ) ({A_{ - 1}},{A_1}) when A − 1 / K ≅ D 4 {A_{ - 1}}/K \cong {D_4} (dihedral group of order 8). There are infinitely many of them, while if A − 1 / K ≅ A 4 {A_{ - 1}}/K \cong {A_4} or S 4 {S_4} there are only finitely many amalgams.