Where $N_G(H)$ is the normalizer of $H$ in $G$ for subgroups $H$ of $G$.
I'd like to have some hints about this question.
I know that $\langle (4 5)\rangle $ is contained in the set.
How can I find other elements using basic group theory? I have seen solutions that use the alternating group but I don't understand them.
Hint: For $g\in S_5$ we have $g^{-1}(1,2,3)g=(1^g,2^g,3^g)$ (where I am using the convention of permutations acting on the right). If $g\in N_G(\langle (1,2,3)\rangle)$ then what can you say about the cyclic decomposition of $g$?
Alternative hint: you noticed $(4,5)\in N_G(\langle(1,2,3)\rangle)$. Which other transpositions are in $N_G(\langle(1,2,3)\rangle)$?
$H=<(1,2,3)>$ is a Sylow 3-subgroup of $S_5$, therefore it is conjugate to all of the other Sylow 3-subgroups in $S_5$. There are $\binom{5}{3}\cdot\frac{1}{2}=10$ such groups. The action of conjugation on $H$ by conjugation has a kernel, namely $N_G(H)$, and $|N_G(H)|\cdot 10=5!=120$. We can conclude that there are $12$ elements in $N_G(H)$.
See if the elements are $e, (1,2), (1,3), (2,3), (1,2,3), (1,3,2), (4,5), (1,2,3)(4,5), (1,3,2)(4,5), (1,2)(4,5), (1,3)(4,5), (2,3)(4,5)$.