Week 9 Problems

These problems are taken from the Chapter 11 homework of the text.

1. (Problem 5) Describe all homomorphisms from $\mathbb{Z}_{24}$ to $\mathbb{Z}_{18}$.

2. (Problem 8) If $G$ is an abelian group, and $n>0$ is an integer, show that $\phi: G\to G$ given by $\phi(g)=g^n$ is a homomorphism. What is the kernel of $\phi$? What about if $G$ is not abelian?

3. (Problem 14) Let $G$ be a finite group and $N$ a normal subgroup of $G$. Suppose that $H$ is a subgroup of $G/N$. If $\phi$ is the canonical homomorphism from $G$ to $G/N$, prove that $\phi^{-1}(H)$ is a subgroup in $G$ of order $|H||N|$.

4. (Problem 16) Suppose $H$ and $K$ are normal subgroups of $G$ and $H\cap K={e}$. Prove that $G$ is isomorphic to a subgroup of $G/H\times G/K$.

5. (Problem 18) Prove that a homomorphism is injective if and only if its kernel consists of the identity element alone.

6. (Problem 19) Given a homomorphism $\phi:G\to H$, define a relation on $G$ by $a\sim b$ if $\phi(a)=\phi(b)$. Show that this is an equivalence relation and describe its equivalence classes.

These problems are not from the text.

1. Let $S$ be the group of roots of unity –that is, complex solutions to $z^n=1$ for some $n$, with multiplication as operation. Prove that $S$ is isomorphic to $\mathbb{Q}/\mathbb{Z}$ by defining a homomorphism $\phi: \mathbb{Q}\to S$ by $\phi(x)=\mathrm{cis}(2\pi x)$, showing that this is surjective, and checking that its kernel is $\mathbb{Z}$.

2. For which groups $H$ does there exist a surjective homomorphism $\phi: D_{6}\to H$? Here as usual $D_{6}$ is the group of symmetries of a regular hexagon.