Find two finite groups $G_1$ and $G_2$ such as:
$1)$ $|G_1|=|G_2|$
$2)$ for all prime $p$ every $p$-sylow subgroup of $G_1$ isomorpic to $p$-sylow subgroup of $G_2$.
but $G_1\not\cong G_2$.
hello I tried to find an example but I couldn't find one please help me. I look on all the groups from order 6 (I choose this order randomly) but I don't know how to prove that every p-sylow subgroup of $G_1$ isomorpic to every p-sylow subgroup of $G_2$ for all primes p.
If $p, q$ are distinct primes then every two groups of order $pq$ will have isomorphic Sylow subgroups (being cyclic subgroups of order $p$ and $q$). So all you have to do is to find two nonisomorphic subgroups of order $pq$, e.g. $\mathbb{Z}_6$ and $S_3$.