91Ó°ÊÓ

Let \(H=\\{I,(12)(34),(13)(24),(14)(23)\\}\). Show that \(H\) is a normal subgroup of \(S_{4}\), so that \(S_{4} / H\) has order six. Write down the six cosets with respect to \(H .\) Is \(S_{4} / H\) is isomorphic to \(C_{6}\) or to \(D_{6}\) ?

Show that \(D_{4}\) is isomorphic to the group \(\\{I, \bar{z},-z,-\bar{z}\\}\) of isometries of \(\mathbb{C}\). This group is known as the Klein 4-group.

Consider the three multiplicative groups $$ \begin{aligned} \mathbb{R}^{+} &=\\{x \in \mathbb{R}: x>0\\} \\ S^{1} &=\\{z \in \mathbb{C}:|z|=1\\} \\ \mathbb{C}^{*} &=\\{z \in \mathbb{C}: z \neq 0\\} \end{aligned} $$ Show that \(\mathbb{R}^{+} \times S^{1}\) is isomorphic to \(\mathbb{C}^{*}\). As \(\mathbb{R}^{+}\)is isomorphic to \(\mathbb{R}\), this shows that \(\mathbb{C}^{*}\) is isomorphic to \(\mathbb{R} \times S^{1}\) (a cylinder).

A set with an associative binary operation with an identity is called a monoid. An element \(x\) in a monoid is a unit if its inverse \(x^{-1}\) exists. Show that the set of units of a monoid \(X\) is a group (even when \(X\) itself is not). We give applications of this idea in the next four exercises.

Show that \(\mathbb{Q} / \mathbb{Z}\) is an infinite group in which every element has finite order.

Let \(H\) be a subgroup of a group \(G\). Show that if \(g_{1} H=H g_{2}\) then \(g_{2} H=H g_{1}\)

Show that the group of units of \(\mathbb{Z}_{8}\) is \(\\{1,3,5,7\\}\). Is this cyclic? Find the group of units of \(\mathbb{Z}_{10}\).

Let \(G\) be a cyclic group of order \(n\) generated by, say, \(g\). Show that if \(k\) divides \(n\), then there is one, and only one, subgroup of \(G\) of order \(k\), and that this is generated by \(g^{n / k}\). [This is a partial converse to Lagrange's theorem.]

Show that every subgroup of rotations in a dihedral group \(D_{n}\) is normal in \(D_{n}\)