Symmetric groups are determined by their character degrees

Let $G$ be a finite group. Let $X_1(G)$ be the first column of the ordinary character table of $G\mathrm{.}$ In this paper, we will show that if $X_1(G)=X_1(S_n),$ then $G\cong S_n\mathrm{.}$ As a consequence, we show that $S_n$ is uniquely determined by the structure of the complex group algebra $\text{C}{S}_n\mathrm{.}$