Symmetric groups are determined by their character degrees

TL;DR

This paper proves that finite groups with the same first column of the character table as a symmetric group are isomorphic to it, establishing that symmetric groups are uniquely determined by their character degrees and complex group algebra structure.

Contribution

It demonstrates that symmetric groups are uniquely identified by their character degree structure and the structure of their complex group algebra, providing a new characterization.

Findings

Finite groups with the same first character table column as S_n are isomorphic to S_n.

Symmetric groups are uniquely determined by their character degrees.

S_n is characterized by the structure of its complex group algebra.

Abstract

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