# Who first noticed the duality for finite groups?

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.

• Dear Sergei, I have no answer (sorry :) but am very interested in the forthcoming feedbacks (+1) – Duchamp Gérard H. E. Mar 3 '19 at 7:57