Homomorphism from unit circle to unit circle In this question $\varphi$ in $\operatorname{Hom}{(S^1, S^1)}$ are of the form $z^n$ all the continuous homomorphisms from $S^1$ to $S^1$ are characterized. However I am looking for non continuous homomorphisms.
My observations are
$S^1$ is isomorphic to $\frac{\mathbb{R}}{\mathbb{Z}}$.
Any subgroup of $\frac{\mathbb{R}}{\mathbb{Z}}$ is in the form $\frac{H}{\mathbb{Z}}$ where $H$ is a subgroup of real numbers containing integers.
If $H$ is generated by $1/n$ then we get $\frac{H}{\mathbb{Z}}$ as the kernel of continuous homomorphisms.
So it is possible to show the existence of non continuous homomorphisms if we can find a non cyclic subgroup $H$ of real numbers such that $\mathbb{R}/H$ is subgroup of $\mathbb{R}/\mathbb{Z}$. But I am not able to find any such H explicitly.
Is there a way to improve this or any alternative ways?
 A: $\mathbb{R}$ is an uncountable-dimensional vector space over $\mathbb{Q}$, so (assuming the axiom of choice) it has an uncountable basis and hence can be written as an uncountable direct sum $\bigoplus_I \mathbb{Q}$. One of these copies of $\mathbb{Q}$ can be chosen to contain $\mathbb{Z}$. The result is that we have an abstract isomorphism
$$\mathbb{R}/\mathbb{Z} \cong \mathbb{Q}/\mathbb{Z} \oplus \bigoplus_J \mathbb{Q}$$
where $J$ is $I$ with one element removed. There are no nonzero homomorphisms from the first summand to the second. This gives
$$\text{End}(\mathbb{R}/\mathbb{Z}) \cong \left[ \begin{array}{cc} \text{End}(\mathbb{Q}/\mathbb{Z}) & \text{Hom}(\bigoplus_J \mathbb{Q}, \mathbb{Q}/\mathbb{Z}) \\ 0 & \text{End}(\bigoplus_J \mathbb{Q}) \end{array} \right].$$
All three of these can be calculated as follows. $\text{End}(\mathbb{Q}/\mathbb{Z})$ turns out to be the profinite integers $\widehat{\mathbb{Z}} \cong \prod_p \mathbb{Z}_p$. $\text{End}(\bigoplus_J \mathbb{Q})$ is an uncountable-dimensional (column-finite) matrix algebra $M_J(\mathbb{Q})$. And $\text{Hom}(\bigoplus_J \mathbb{Q}, \mathbb{Q}/\mathbb{Z}) \cong \prod_J \text{Hom}(\mathbb{Q}, \mathbb{Q}/\mathbb{Z})$. $\text{Hom}(\mathbb{Q}, \mathbb{Q}/\mathbb{Z})$ is a bit tricky to describe but you can see here that one possible description is $\mathbb{Q} \otimes \widehat{\mathbb{Z}}$, and you can find another description working one prime at a time towards the end of these notes.
So this is a more or less complete description of all endomorphisms, nearly all of which are highly discontinuous.
