# HW, 3rd exercise

Greetings, about the third exercise, I have to make a comment here, if I’m not mistaken, we need to choose $x\neq 0$ because $0$ also have the property that $\Delta(0)=0\otimes0$ but it doesn’t belong to the group.
I think that could help. (Thanks to Yiby Morales, she encouraged me to post this!)

1. Brian Cruz

Thanks Sebastian (and thanks Yiby)!

I was thinking the same thing and came to the forum to see if anyone had posted about it! I think you’re right, and the PlanetMath website seems to say the same thing about the grouplike element.

http://planetmath.org/encyclopedia/GrouplikeElementsInHopfAlgebras.html

2. alyssapalfreyman

In the group ring is it appropriate to consider the elements of $latexG$ as a basis for $latex\mathbb{F}[G]$ and therefore the pure tensors of elements of $latexG$ as a basis for $latex\mathbb{F}[G]\otimes\mathbb{F}[G]$? My hope is that two elements of $latex\mathbb{F}[G]\otimes\mathbb{F}[G]$ are equal only if the coefficients of corresponding pure tensors are equal. Is this true?

3. alyssapalfreyman

also, how do you get the latex to compile?

4. Sebastián O.

alyssapalfreyman, the basis of the $\mathbb{F}[G] \otimes \mathbb{F}[G]$ are exactly the elements $g \otimes g'$ with $g, g' \in G$

PD: You write $, then latex, spacebar, then the latex code, and last you close with$.