Proving Permutation Inverses: (1 2 3) = (4 5 6)

[Justabeginner]

Yes, that's correct. Note that this solution changes the cycle (1 2 3) to (4 5 6), and it changes the singleton cycles (4), (5), (6) to (1), (2), (3) respectively. In other words, writing out the full cycle notation including singletons:
$\sigma$(1 2 3)(4)(5)(6)$\sigma^{-1}$ = (4 5 6)(1)(2)(3)

There are five other solutions in which (4), (5), and (6) are mapped to different arrangements of (1), (2), and (3), for example:
$\sigma$(1 2 3)(4)(5)(6)$\sigma^{-1}$ = (4 5 6)(1)(3)(2)

The right hand side (4 5 6)(1)(3)(2) is the same permutation as (4 5 6)(1)(2)(3), since disjoint cycles commute, but we obtain it using a different $\sigma$.

If you're bored, you can try to find the other solutions. For example, to transform
(1 2 3)(4)(5)(6) to
(4 5 6)(1)(3)(2),

$\sigma$ needs to map as follows:
$1 \mapsto 4$
$4 \mapsto 1$
$2 \mapsto 5$
$5 \mapsto 3$
$3 \mapsto 6$
$6 \mapsto 2$
and therefore $\sigma$ = (1 4)(2 5 3 6) is another solution to the problem.

I did not know about singletons; thank you for that information!