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A class of finite group-amalgams


Author: Dragomir Ž. Djoković
Journal: Proc. Amer. Math. Soc. 80 (1980), 22-26
MSC: Primary 20E99
MathSciNet review: 574502
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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.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1980-0574502-9
Article copyright: © Copyright 1980 American Mathematical Society