If $G$ is a finite group of order n then $G$ is isomorphic to a subgroup of the alternating group $A_{n+2}$.
I found this in a book GROUPS AND SYMMETRY by M. Armstrong. I can prove this by extending a cycle of order $n$ to a cycle of order $n+2$ in a unique way.
But my question is Is this also true for $A_n$?
can't find any suitable example to disprove my statement.
No, the cyclic group $C_6$ is not a subgroup of $A_6$. All subgroups of order $6$ in $A_6$ are isomorphic to $S_3$. In fact, the cycle $(123456)$ is not in $A_6$.