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.