Consider the set $F$ of all functions from $\{1,2,3\}$ to $\{1,2,3\}$. There are $3^3= 27$ of them.
Prove this set is not a group under function composition.
I thought that it violates the inverse element property, but not sure how. I believe identity in our case is the identity map. Not really sure how to show an example how it fails under inverse. Help much appreciated
You had the right idea. Consider this: Is there an inverse of the constant function $f$ defined by $f(1)=f(2)=f(3)=1$?