Requirements to have a group consisting of a set of functions and function composition?

Question

I apologise if this is an elementary qiestion. I am teaching myself group theory and am attempting some questions which i am finding rather tricky. In one question, I am asked to show that function composition is associative and then what extra properties are needed to get a group with composition as the group operation.

So considering the group axioms, function composition is associative, so that takes care of the requirement of associativity. The set of functions must also include the identity map (existence of identity element).

Now considering what other requirements there may be, I thought that for the set of functions to be comparable they must all be maps of $X \to X$ and must all be bijections (needed for the existence of inverse too I think).

I am trying to find out if it is indeed true that all the functions in a set in a group need be bijections, but I have not come across this being stated. I was hoping someone could verify if my ideas are correct.

Answer 1

Group axioms force your function to have a left and right inverse and, moreover, they must be equal. This implies that all the elements of your group are necessarily bijections.

Answer 2

There are groups of functions where not all of them are bijections, but they might as well be!

The Linear Fractional Transformations of the Complex Plane are not all bijective mappings.