# No Eckmann-Hilton

Prove the following two statements and conclude that the fundamental group of a (connected) topological group $G$ is abelian:

1) A discrete normal subgroup of a connected topological group is central.
2) If $p:\hat{G}\to G$ is the universal covering, the total space $\hat{G}$ admits a topological group structure such that $p$ is a group morphism. The kernel of $p$ is then isomorphic to the fundamental group of $G$.

# Rank isn’t like dimension

Let $F_2(a,b)$ be the free group in the generators $a$ and $b$. Consider the group morphism $f: F_2(a,b) \to \mathbb{Z}$ defined by $f(a)=f(b)=1$. Prove that $Ker(f)$ is a free group of infinite rank. (Hint: Think topologically!)

# Outside in

Given a group $G$, embed $G$ in a larger group $H$ such that any automorphism of $G$ is the restriction of an inner automorphism of $H$.

# Spooky algebra II

Exhibit a group $G$, a set of generators $X$ for $G$ and a subgroup $H$ such that $xHx^{-1} \subseteq H$ for all $x\in X$ and yet $H$ is not normal in $G$.

# On primitive roots of unity

Finite subgroups of the group of units of a field are cyclic.

# Freedom

Every morphism $f:\Bbb{Z}^\Bbb{N}\to A$ of abelian groups vanishing on $\Bbb{Z}^{(\Bbb{N})}$ is identically zero.

Use this fact to prove that $\Bbb{Z}^\Bbb{N}$ is not a free abelian group.

# Group theory is hard

Let $G$ be a finite group such that $G/Z(G)\simeq \Bbb{Z}/p\Bbb{Z}\oplus\Bbb{Z}/p\Bbb{Z}$, with $p$ prime. Then $p$ divides the order of $Z(G)$.