Three questions on group algebras

• How is the group embedded in the group algebra?

Let $K$ be a field, $G$ a group and $A=K[G]$ the group algebra. In Magma, for a group algebra A and an element g of the group, writing A!g gives the element $1_Kg\in A$. You can also recover G with Group(A).

• Can we compute (some) units of a group algebra?

If the group algebra is finite, the best way I can find to do this is looping through the elements of the group algebra and then asking if they are units or not; this is done with the IsUnit function. If the group algebra is not finite, then I don’t know yet, but you can certainly still test whether specific elements are units or not with the same function.

• Can you compute (some) idempotents of a group algebra?

Similar to the previous point, but with the function IsIdempotent.