Definition of “Grouplike”

Hi!  In terms of a nonzero element x\in \mathbb{F} [G] being defined as grouplike,is it sufficient to have one comultiplication such that

\Delta (x) = x \otimes x or does it require that to be true for every comultiplication on the space? Thank you.

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One comment

  1. It should satisfy \Delta(x) = x\otimes x with \Delta being the canonical comultiplication (i. e., the one defined by \Delta(g) = g\otimes g for every g \in G . All the other comultiplications don’t matter.

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