Just infinite group A group is said to be just infinite if it is infinite and every proper quotient is finite.
In this paper  http://onlinelibrary.wiley.com/doi/10.1002/cpa.3160210604/pdf I read that infinite dihedral group $D_\infty$ is just infinite and its proper quotients are dihedral groups of type $D_{2n}$.
Why is this?
I ask also if is there a name for the dual notion: a group infinite with every not trivial (normal?) subgroup finite.
For exemple $\mathbb{Z}(p^\infty)$ and Tarsky group have this property; are there any other interesting examples?
 A: The infinite dihedral group acts faithfully on the line $\mathbb{R}$ by reflections and translations. Since the line is a tree, and for any group acting on a tree, any finite order element fixes a point, we can conclude that all finite order elements of $D_\infty$ are either trivial, or reflections. Now suppose a subgroup contains two non-trivial finite order elements, these must be distinct reflections, and hence generate a subgroup isomorphic to $D_\infty$. On the other hand, suppose that a subgroup contains no non-trivial finite order elements, then it consists of just translations, and is isomorphic to $\mathbb{Z}$. Putting all this together, a complete list of subgroup isomorphism types for $D_\infty$ is $$\{1\},\;\mathbb{Z}_2,\;\mathbb{Z},\;\textrm{and}\;D_\infty.$$ You can check that the normal closure of any copy of $\mathbb{Z}_2$ in $D_\infty$ is a copy of $D_\infty$ of index 2, and all other non-trivial subgroups have finite index, so their normal closures do too.
I don't know off the top of my head an answer to your second question, sorry.
