A class of finite group-amalgams

Abstract:

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