TITLE: What objects can belong in a group?
QUESTION [8 upvotes]: 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!

REPLY [6 votes]: Any collection of objects can be made into a group. For example, the set consisting of a Teddy bear and an apple can be made into a group as follows:
Let $G$ be the set $\{$ ʕ•͡ᴥ•ʔ,  $\}$. Define a binary operation $+: G \times G \to G$ as follows:
ʕ•͡ᴥ•ʔ $+$  $=$  $+$ ʕ•͡ᴥ•ʔ $=$ ʕ•͡ᴥ•ʔ
and
ʕ•͡ᴥ•ʔ $+$ ʕ•͡ᴥ•ʔ $=$ 
and 
 $+$  $=$ 
The definition of a group is merely a blue print that you can use to spot objects that are groups. That elements of groups are denoted by integers and other collections of numbers is just for convenience:
One reason is that it is easier to write $1$ than it is to write ʕ•͡ᴥ•ʔ. 
Another reason is that we expect $1$ and $-1$ etc. to behave in certain familiar ways and the notation delivers this intuition to us at no additional cost. 
But these two reasons don't mean that we couldn't write $\mathbb Z / 6 \mathbb Z$ entirely using funny symbols of our own choice.