Classification of locally dihedral amalgams. (Q863375)

The author studies amalgams \((A_1,A_2)\) of the following type: \(A_1/K\) is a dihedral subgroup of order \(2k\) and \(|A_2:A_1\cap A_2|=2\). Here \(K\) is the largest subgroup of \(A_1\cap A_2\) which is normal in \(A_1\). As usual one assumes that no nontrivial subgroup of \(A_1\cap A_2\) is normal in \(\langle A_1,A_2\rangle\). For \(k=3\) these amalgams have been classified by \textit{D. Ž. Đoković} and \textit{G. L. Miller} [J. Comb. Theory, Ser. B 29, 195-230 (1980; Zbl 0385.05040)] and for odd \(k\) in general by the author [J. Algebra 298, No. 2, 630-644 (2006; Zbl 1098.20024)]. For \(k=4\) there are some results due to \textit{D. Ž. Đoković} [Proc. Am. Math. Soc. 80, 22-26 (1980; Zbl 0441.20015)]. In this paper the author gives a complete classification for \(k\) even in terms of generators and relations, to complicated to be stated here.