= Complex group algebra detects p-nilpotency
The matrix-block sizes in the <Artin–Wedderburn theorem> for $\mathbb C[G]$ recover the <irreducible character degrees> with their multiplicities. They determine the degree-square sum and the number $|G:G'|$ of linear characters. <Isaacs' character-degree criterion for p-nilpotency> therefore detects each normal p-complement and applying it for every prime also detects <nilpotent groups>. The set of distinct degrees without multiplicities does not contain the same information.
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