Group algebras seem to be buggy
I want to work with group algebras in SageMath. But they do not behave as they should. Is this a bug or do I do something wrong? For my application, I need to work with a group defined by a presentation. So the simplest example should be $\mathbb{F}_2[C_2]$, with $C_2 = \langle X : X^2 = 1\rangle$.
sage: F.<X> = FreeGroup(1)
sage: G = F / [X^2]
sage: display(G)
Finitely presented group < X | X^2 >
sage: A = G.algebra(GF(2))
sage: x = A.gen()
sage: x^2 == A.one()
False
It seems that the relations of $G$ will not be presented in $R[G]$. Why?
Let's try to define the group as a permutation group instead. This throws an error.
sage: G = PermutationGroup([[2,1]])
sage: A = G.algebra(GF(2))
sage: x = A.gen()
sage: x^2
()
sage: A.one()
()
sage: x^2 == A.one()
RuntimeError: There is a bug in the coercion code in Sage.
There is also an error when I just want to compute $(x+1)^2$.
PS: SageMath seems to be so buggy that I consider switching to a different CAS.
The second one works for me in sage 10.2.beta5. Do you use an old version of sage ?
Thank you. This may be the reason. I use Sage version 9.5. The Sage Cell Server probably uses a different version, and there the 2nd code snippet works. (Still weird that the 1st doesn't work.)
First one is already failing at the level of the group quotient, yes.