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Symmetric groups are determined by their character degrees. (English) Zbl 1246.20007
In this short paper the author proves the unsurprising result stated in the title: that if \(G\) is a finite group and the degrees of the irreducible characters of \(G\) (with multiplicities) are the same as those of the symmetric group \(\mathfrak S_n\), then \(G\cong\mathfrak S_n\). It is not clear why this result might be interesting, and the author’s rather laboured introduction sheds no light on this.
The proof is well presented, but involves a great deal of imported material, including the classification of finite simple groups, and small character degrees of the sporadic groups. It would be good to see a more elementary argument.

MSC:
20C15 Ordinary representations and characters
20C30 Representations of finite symmetric groups
20D60 Arithmetic and combinatorial problems involving abstract finite groups
Software:
GAP
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References:
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