Background. Let $G$ be the automorphism group of a string $s$, ie. $G = \langle (i,j) : i \lt j, s[i] = s[j]\rangle$. Then $G$ is a normal subgroup of $S_{|s|}$ the symmetry group on $|s|$ symbols.
Proof. $G \approx S_{k_1} \times \cdots \times S_{k_n} = H_1 \cdot H_2 \cdots H_n$ where $H_i = 1 \times \dots \times 1\times S_{k_i} \times 1 \times \dots \times 1$ each of which is isomorphic to $S_{k_i}$. Each such copy is normal as $S_{k_i}$ is the symmetric group on $k_i$ symbols and contains every transposition on $k_i$ symbols. In other words use that to prove $\tau S_{k_i} = S_{k_i}\tau, \ \ \forall $ transpositions $\tau \in S_{|s|}$.
Also, there is a homomorphism from $S_{|s|}$ onto $G$ given my multiplying all the projections together.
Are these correct proofs?