Subgroup of translation group I am trying to understand factor (quotient) groups and I thought maybe constructing my own example would help me.
Let $H_{\lambda} = \lambda \mathbb{Z}$ with $\lambda \in \mathbb{R}$ be a subgroup of the the 1-dimensional translation group $T$. Since $T$ is abelian then $H_\lambda$ is an invariant subgroup. Then consider the factor group $K_\lambda = T/H_\lambda$.
Does $K_\lambda$ basically contain actions that say translate by $(x+m\lambda)$ for $x \in T$ and $m \in \mathbb{Z}$ or is that conclusion incorrect?
I am concluding this because the elements of $K_\lambda$ are $xH_{\lambda}$ for $x \in T$. 
 A: I think you may have some confusion around what the elements of $K_{\lambda}$ are. 
A quotient (factor) group $G/N$ of a group $G$ by a normal subgroup $N$ of $G$ is by definition a bunch of certain subsets of $G$, called cosets, endowed with a group operation. These cosets are defined as the sets $xN = \{ xn : n \in N \}$. 
You can show that for any subgroup $H$ (not necessarily normal) that these cosets form a partition of $G$: for any $x,y \in G$, either $xH = yH$ or $xH \cap yH = \emptyset$, and $G$ is the union of all cosets. When a subgroup $N$ is normal, by defining a product on these subsets by $xN \cdot yN = xyN$, these subsets form a group.
Saying “Does $K_{\lambda}$ basically contain actions” doesn’t really make sense: $K_{\lambda}$ contains collections of actions. It is true however, that each subset of actions in $K_{\lambda}$ can be described as $\{ x + m\lambda \}_{m \in \mathbb{Z}}$ for a fixed $x$. This is because two elements $a,b$ are in the same coset if and only if $a - b \in N$.
