91Ó°ÊÓ

Find the order of the indicated element in the indicated quotient group. $$ 2+\langle 6\rangle \text { in } \mathbb{Z}_{15} /\langle 6\rangle $$

In Exercises 1 through \(10,\) determine whether or not the indicated map \(\phi\) is a homomorphism, and in the cases where \(\phi\) is a homomorphism, determine Kem \(\phi\). $$ \phi: \mathbb{Z}_{7} \rightarrow \mathbb{Z}_{2}, \text { where } \phi(x)=\text { the remainder of } x \text { mod } 2 $$

Let \(H\) be a subgroup of a group \(G\). Show that for any \(a \in G\) we have \(|H a|=|H|\).

For \(p\) a prime show that \(\operatorname{Aut}\left(\mathbb{Z}_{p}\right) \cong \mathbb{Z}_{p-1}\).

Determine whether the indicated subgroup is normal in the indicated group. Find all the normal subgroups in \(\mathrm{GL}\left(2, \mathbb{Z}_{2}\right)\), the general linear group of \(2 \times 2\) matrices with entries from \(\mathbb{Z}_{2}\).

Let \(Q_{8}\) be the quarternion group. Show that \(\operatorname{Inn}\left(Q_{8}\right)=V,\) the Klein 4 -group.

Let \(H\) be a subgroup of a group \(G\). Show for any \(a \in G\) that \(a H=H\) if and only if \(a \in H .\)

Determine whether the indicated subgroup is normal in the indicated group. For \(r \in \mathbb{R}^{*}\) let \(r I=\left[\begin{array}{rr}r & 0 \\ 0 & r\end{array}\right]\). Show that \(H=\left\\{r l \mid r \in \mathbb{R}^{*}\right\\}\) is a normal subgroup of \(\mathrm{GL}(2, \mathbb{R})\)

Show that \(\mid\) Aut \(\left(D_{4}\right) \mid \leq 8\).

For \(V\) the Klein 4 -group show that \(\operatorname{Aut}(V) \cong \mathrm{GL}\left(2, \mathbb{Z}_{2}\right),\) the general linear group of \(2 \times 2\) matrices with entries from \(\mathbb{Z}_{2}\).