数学代写|Abstract Algebra代考

Q1
Let $n \geq 1$ be arbitrary. Consider the group $G=\operatorname{Aut}\left(\mathbb{Z}_n\right)$.
(a) Find a group we have seen in this course isomorphic to $G$.
(b) Let $F: G \rightarrow \mathbb{Z}_n$ be the function defined by $F(g)=g(1)$. Explain why $F$ is injective.
(c) Let $H \subseteq \mathbb{Z}_n$ be the image of $F(G)$. Explain why $H$ is not a subgroup of $\mathbb{Z}_n$.
(d) Find an example where $H$ is isomorphic to a subgroup of $\mathbb{Z}_n$.
(d) Find an example where $\operatorname{Aut}\left(\mathbb{Z}_n\right)$ is isomorphic to $\operatorname{Aut}\left(\mathbb{Z}_m\right)$ but $m \neq n$.

Q2
Consider the group $G=\mathbb{Z}^2$.
(a) Prove that $H={(3 m, 2 n): m, n \in \mathbb{Z}}$ is a subgroup.
(b) Find two representative elements from each of the cosets of $H$.
(c) Plot the elements of $\mathbb{Z}^2$ as points in the plane. Colour each of the cosets of $H$ in a different colour.
(d) Describe a group isomorphism from $G$ to $H$.
(e) Is this an automorphism? Explain.

Q3
Consider $D_6$ the group of symmetries of a regular hexagon.
(a) Label the vertices of the hexagon with the numbers $0,1, \ldots, 5$. Explain why $D_6$ is isomorphic to a subgroup of $S_6$.
(b) Use Lagrange’s Theorem to find the number of cosets of $D_6$ in $S_6$.
(c) Consider the subgroup $K=S t a b_{S_6}(1)$ of $S_6$. Determine the order of $K$.
(d) Determine the number of elements in the set $D_6 K$.
(e) Explain why $D_6 K$ is a subgroup of $S_6