Proposition. There exists a group generated by two elements with uncountably many normal subgroups.
The constructive proof below appears in Geometric Group Theory: An Introduction by Clara Löh (Proposition $2.2.30$).
Proof. The basic idea is as follows: We construct a group $G$ generated by two elements that contains a central subgroup $C$ (i.e., each element of this subgroup is fixed under conjugation by all other group elements) isomorphic to the big additive group $\bigoplus_{\Bbb N} \Bbb Z$. The group $C$ contains uncountably many subgroups (e.g., given by taking subgroups generated by the subsystem of the unit vectors corresponding to different subsets of $\Bbb N$), and all these subgroups of $C$ are normal in $G$ because $C$ is central in $G$.
- What is the explanation for why $\bigoplus_{\Bbb N} \Bbb Z$ has uncountable many subgroups? I don't understand the stuff about the "subsystem of unit vectors" corresponding to subsets of $\Bbb N$.
An example of such a group is $G := \langle s,t \, \vert \, R\rangle$, where $$R := \{[[s,t^nst^{-n}],s]: n \in \Bbb Z\} \cup \{[[s,t^nst^{-n}],t]: n \in \Bbb Z\}$$ Let $C$ be the subgroup of $G$ generated by the set $\{[s,t^nst^{-n}]]: n \in \Bbb Z\}$. All elements of $C$ are invariant under conjugation with $s$ by the first part of the relations, and they are invariant under conjugation with $t$ by the second part of the relations; thus, $C$ is central in $G$. Moreover, using the so-called calculus of commutators, it can be shown that $C$ contains the additive group $\bigoplus_{\Bbb N} \Bbb Z$.
- For the last part, we must construct an injective group homomorphism $\varphi: \bigoplus_{\Bbb N} \Bbb Z \to C$. It is enough to define $\varphi$ on a generating set of $\bigoplus_{\Bbb N} \Bbb Z$, for which the ideal pick is $S := \{e_n: n \in \Bbb N\}$ where $e_n = (0, \ldots, 0, 1, 0, \ldots)$ has a $1$ in the $n$th coordinate and $0$s elsewhere for each $n\ge 1$. The relations in $G$ can be rewritten as $[st^nst^{-n}s^{-1}t^ns^{-1}t^{-n},s] = e_G$ and $[st^nst^{-n}s^{-1}t^ns^{-1}t^{-n},t] = e_G$ for all $n \in \Bbb Z$ where $e_G \in G$ is the identity element. How should I define $\varphi$? I don't see why the commutators $[[s,t^nst^{-n}],s]$ and $[[s,t^nst^{-n}],t]$ are useful to consider in the first place.
Edit: As in Derek's answer, I am tempted to map generators to generators, i.e., define $\varphi(e_n) = [s, t^n s t^{-n}]$ for all $n \in \Bbb N$. However, I haven't been able to check if $\varphi$ is injective.
Thanks for any help!
Question 1. $C \cong \bigoplus_{n \in {\mathbb N}} {\mathbb{Z}}$ is generated by elements $e_i$ with $i \in {\mathbb N}$. For each of the uncountably many subsets $J$ of ${\mathbb N}$ we can define the subgroup $\langle e_j: j \in J \rangle$ of $C$.
Question 2. For $n \in {\mathbb N}$, define $\phi(e_n) = [s,t^n s t^{-n}]$. The commutators are needed to ensure that the subgroup $C$ lies in the center of $G$. To make $\phi$ an isomorphism just redefine $C$ to be the image of $\phi$, i.e. the subgroup generated by $[s,t^n s t^{-n}]$ for $n \in {\mathbb N}$.