A.A.Kirillov in section 12.3 of his "Elements of the Theory of Representations" writes that the first "symmetric" duality theory for non-commutative groups was the theory for finite groups. In short what he writes can be explained as follows: the operation $G\mapsto{\mathbb C}_G$ that assigns to each finite group $G$ its group algebra ${\mathbb C}_G$ over ${\mathbb C}$, embeds the category of all finite groups into the category of finite dimensional Hopf algebras, so that the Pontryagin duality functor $G\mapsto\widehat{G}$ (for Abelian finite groups) turns into the operation $H\mapsto H^{*}$ of taking the dual vector space (which is a duality functor in the category of finite dimensional Hopf algebras).

This observation is used as a key example in different modern duality theories. I wonder

who first noticed this?

Kirillov does not give the name. Duality for finite groups.

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    $\begingroup$ Dear Sergei, I have no answer (sorry :) but am very interested in the forthcoming feedbacks (+1) $\endgroup$ – Duchamp Gérard H. E. Mar 3 at 7:57

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