is there any example of a group of order 4 which is not cyclic.

Question

there is no concrete example in the book.

Answer 1

The product of two groups of order 2. ($Z/2Z\times Z/2Z)$. In this group, every element is of order 2 so it is not a cyclic group of order 4.

Answer 2

Its the Klein four-group $V=\{e,a,b,c\}$ which is isomorphic to ${\Bbb Z}_2\times {\Bbb Z}_2$ (additively) and so not cyclic. In this group, if $e$ is the unit element, then $a^2=b^2=c^2=e$.