To construct a non-abelian group of order $55$ and $203$
Solution 1:
Yes, this is correct.
In general, if $p$ and $q$ are primes with $p$ dividing $q-1$, then you can find an automorphism of the cyclic group $C_q$ of order $p$, and then form the corresponding semidirect product of $C_p$ by $C_q$ to obtain a non-abelian group of order $pq$.