Suppose $G$ is the group of all functions between $[0,1]\to\mathbb{Z}$. Let $H$ be the subgroup defined as $H=\{f\in G: f(0)=0\}$. Then, what can be said about the cardinality of $H$ and its index in $G$?
I think the cardinality of $H=G=2^{c^2}$, where $c$ is the cardinality of the natural numbers. Hence, I think the index is countable. Am I right? Any hints ? Thanks beforehand.
Hint. Prove that $G/H\simeq\mathbb{Z}$ by exhibiting a surjective group morphism $G\to \mathbb{Z}$ with kernel $H$ (there is an obvious one).
Then deduce that $G/H$ has cardinality $\aleph_0$ ($c$, with your weird notation).