intersection of sylow subgroup I am in need of 


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*Example of a group $G$ a subgroup $A$ which not normal in $G$ and $p$ sylow subgoup $B$ of $G$ such that $A \cap B$ is a $p$-sylow subgroup of $A$.


A detailed solution will be helpful.

If you want to construct an example which is non-trivial under several points of view, that is $A$ and $B$ and $G$ all distinct, and $A$ is not a $p$-group, you may find it as follows.
Find a finite group $G$ which has a non-normal subgroup $A$ which is not a $p$-group, but whose order is divisible by $p$. Let $S$ be a Sylow $p$-subgroup of $A$. Choose $B$ to be a Sylow $p$-subgroup of $G$ containing $S$. (Such a $B$ exists by Sylow's theorems.)
For instance, take $p = 2$, $G = S_{4}$, $A = \langle (123), (13) \rangle$ of order 6, $S = \langle (13) \rangle$, $B = \langle (1234), (13) \rangle$.
PS A simpler, but somewhat trivial example would be $G = B = \langle (1234), (13) \rangle$, $A = \langle (13) \rangle$.
