Math 120A HW 10

1. Let $G$ be a group and let $H$ be a normal subgroup of $G$.
   1. Prove that if $G$ is abelian, then the factor group $G/H$ is abelian.
   2. Give an example in which $G$ is not abelian, but $G/H$ and $H$ are both abelian.
   3. Explain why your example shows that $(G/H) \times H$ is not always isomorphic to $G$.
2. Let $G$ be a group and let $H$ be a normal subgroup of $G$. Let $m = (G : H)$, meaning the index of $H$ in $G$. Prove that $a^m \in H$ for all $a \in G$. Hint: consider the orders of elements in a factor group.
3. Let $n \ge 2$. We know that the factor group $S_n/A_n$ must be isomorphic to $\mathbb{Z}_2$ because it is a group of order $2$ and every group of order $2$ is isomorphic to $\mathbb{Z}_2$, but this is not very satisfying. Show $S_n/A_n \cong \mathbb{Z}_2$ directly by defining a function from $S_n/A_n$ to $\mathbb{Z}_2$ and showing that it is a (well-defined) isomorphism.
4. Let $H$ be the cyclic subgroup of $\mathbb{Z} \times \mathbb{Z}$ generated by $(1,1)$.
   1. Give a condition on the numbers $i,j,k,l \in \mathbb{Z}$ that is necessary and sufficient for the two cosets $(i,j) + H$ and $(k,l) + H$ to be equal. Your condition should not mention $H$.
   2. Show that the coset $(0,1) + H$ is a generator of the factor group $(\mathbb{Z} \times \mathbb{Z}) / H$.
   3. Show that the coset $(0,1) + H$ has infinite order in the factor group $(\mathbb{Z} \times \mathbb{Z}) / H$. (It is not enough to show that the element $(0,1)$ has infinite order in $\mathbb{Z} \times \mathbb{Z}$.)
   Hint: for parts b and c, you can use the general fact that $n(a +H) = (na) + H$ when $a$ is an element of an abelian group, $H$ is a subgroup, and $n$ is an integer. When $n = 2$ this fact follows from the definition of $(a+H) + (a+H)$, and the proof of the general case uses induction.
5. Let $G_1$ and $G_2$ be groups, let $H_1$ be a normal subgroup of $G_1$, and let $H_2$ be a normal subgroup of $G_2$. Prove that $$ (G_1 \times G_2)/(H_1 \times H_2) \cong (G_1 / H_1) \times (G_2 / H_2).$$ Hint: what does an element of the left hand side look like in terms of elements $a_1,a_2 \in G$? How about the right hand side? What is a natural function that takes one to the other? Prove that it is a (well-defined) isomorphism.
6. Compute the following factor groups. (To "compute" a factor group means to find a more familiar group that is isomorphic to it. More precisely: in this case the factor groups are finite abelian groups, so you can classify them according to the fundamental theorem of finite abelian groups.)
   1. $(\mathbb{Z}_4 \times \mathbb{Z}_4) / \langle (1,2) \rangle$
   2. $(\mathbb{Z}_4 \times \mathbb{Z}_4) / \langle (2,2) \rangle$
   3. $(\mathbb{Z}_4 \times \mathbb{Z}_4) / \{(0,0),(0,2),(2,0),(2,2)\}$
   Hint: In each case, what is the order of the factor group? What are the possibilities (up to isomorphism) for an abelian group of that order? How can you distinguish between the possibilities?