Answer to Question #308432 in Abstract Algebra for Chinmoy kumar Bera

Question

the set of all functions of {1 2 3...,10} to itself forms a group w.r.t the composition of function.true or false

Answer 1

Consider the function

$f:\{1, 2 ,3,...,10\}\to\{1 ,2, 3,...,10\},$

$f(x)=1.$

This function is not a bijection, and hence is not invertible. Therefore, $f$ has no inverse, and we conclude that the set of all such functions does not form a group with respect to the operation of composition.