Q:

Naming a "split" subgroup of the symmetric group?

Let $a+b = n$. I'm interested in the group that permutes first $a$ elements then the next $b$ elements. It should be a subgroup of the symmetric group $S_n$, right?
For example, $a=1,b=2$, the group should have elements $\{(1)(2)(3),(1)(2,3), (1)(3,2)\}$.
What do we call such a permutation group? 
Given such a group $G$, how do we express the sets which the groups permutes, e.g. $\{1\}, \{2,3\}$. These are ...(something?)... of the group $G$, cosets? 
I know I'm not 100% clear, but I don't know how to express myself better.

A:

The group you describe is indeed a subgroup of $S_n$, and it is isomorphic to $S_a\times S_b$. The symmetric group $S_n$ acts naturally on the set $\{1,\ldots,n\}$ and the corresponding subgroups $S_a$ and $S_b$ act naturally on the sets $\{1,\ldots,a\}$ and $\{a+1,\ldots,n\}$. These sets are the orbits of the action of $S_a\times S_b$ on the set $\{1,\ldots,n\}$.

