What objects can belong in a group? 
What type of object can belong in a group?

So far I've only seen integers and matrices. Can vectors be in a group? Can operators be in a group?
Someone please help stretch my mind a bit as online notes does not help in this respect. Thanks!

Any object at all can be in a group. What makes a set a group is the operations with which one endows it, namely, the multiplication map $$G \times G \to G$$
and inverse map $$G \to G$$
satisfying the usual group axioms.
For example, the vector space axioms imply that for any vector space $(\mathbb{V}, +, \cdot)$, the underlying set $\mathbb{V}$ is an (abelian) group under the given addition operation $+$.
And of course, we can think of elements of a matrix group (for example) $G \subseteq GL(n, \mathbb{R})$ as unary operators that act on $\mathbb{R}^n$, e.g., by left multiplication.
