A question on $G=S_5$ concerning notation

Question

I think there must be a misprint in a question I've been asked.It has written on it ;

$$|G|=S_5$$

but of course this notation is nonsense as a real number cant be equal to a set.

Do you think it was supposed to read

i)$|G|=|S_5|$

ii)$G=S_5$

I also have the additional question that if $|G|=|S_5|$ then can we assume that G is isomorphic to $S_5$ and still consider the elements of G to be permutation cycles ?

Answer 1

It is probably like you wrote it, i.e $G = \mathcal{S}_5$.

$|G| = |\mathcal{S}_5|$ doesn't imply that $G = \mathcal{S}_5$, since there are 47 groups of order 120 up to isomorphism.