# HW #3, 4(e)

I’m trying to calculate the order of the antipode in 4(e), and I keep coming up with the same result as in part (c).  Conceptually, I don’t see how this can be otherwise, since all we’ve done is impose some relations on the bialgebra.  Since modding out preserves equality, then if $S^k(x)=x$ in $H$, then we should have $S^k(x)/\{\text{relations}\}=x/\{\text{relations}\}$ in $H/\{\text{relations}\}$.  This implies that the order of $S$ in $H/\{\text{relations}\}$ should be less than or equal to the order of $S$ in $H$.  However, problem 4(e) implies that we can make the order of $S$ as large as we like in $H/\{\text{relations}\}$ by picking a sufficiently large $n$.  Anyone feel like weighing in?

In my proof, I used what I had in part (c) to prove part (e). My suggestion to you would be to look at what you have for part (c) and then think about how you can use *part* of that to conclude that the antipode of $J$ must have even order, for some given $n$ as indicated in the problem.