Chapter 13

Show that the dihedral group \(D(4)\) is not an internal direct product of any two of its proper subgroups.

Prove that the group with presentation $$ \left\langle x, y: x^{15}=1=y^{2}, y x y^{-1}=x^{4}\right\rangle $$ is isomorphic to the direct product \(C_{3} \times D(5)\).

Show that the dihedral group \(D(6)\) is an internal direct product of a subgroup of order 2 with a subgroup of order 6 isomorphic to the dihedral group \(D(3)\).

Complete the calculation of the number of elements of order 2 in the central product of two copies of the group \(D(4)\). Find how many elements of order 2 there are in the central product of two copies of the quaternion group of order eight.

Let \(G\) be a cyclic group of order 6 generated by \(x\) and \(H\) be the alternating group \(A(4)\), with \(N\) equal to \(\left\langle x^{3}\right\rangle\) and \(K\) equal to the subgroup \(\\{1,(12)(34),(13)(24),(14)(23)\\}\). Define a map \(\vartheta: G / N \rightarrow H / K\) by $$ \vartheta(N)=K, \quad \vartheta(x N)=(123) K \quad \text { and } \vartheta\left(x^{2} N\right)=\left(\begin{array}{lll} 13 & 2 \end{array}\right) K $$ List the elements in the pullback of \(G\) and \(H\) via \(\vartheta\).