Determine the group of symmetry of the pictures, giving a set of generators. Are these groups isomorphics?
First I determined the group of symmetries, $J=\{I_2,A_{\pi/2},A_\pi,A_{3\pi/2}\}= <A_{\pi/2}> $
for the picture in the left side and $G =\{I_2,A_\pi,F_v,F_h\} = <A_\pi,F_v>$ for the picture in the right side where $F_v$ represents a vertical reflection and $F_h$ an horizontal reflection. Then I made a Cayley Table:
I observed the diagonals are diferents then the groups aren't isomorphics. Is this correct? Is there another way to solve this problem?
Yes this is fine. That the diagonals can't be made to match up indeed shows that the groups are not isomorphic. Another way (really basically the same) is to observe that one group is cyclic of order 4, while in the other group each non-identity element has order 2.
If you haven't done it already, as an exercise show that every group of order 4 is isomorphic to one of these two groups. (This can likely be found on MSE.)