Let $V(FA_4)$ be normalized unit group of group algebra $FA_4$, where $F$ is a field containing 4 elements and $A_4$ is alternating group on $4$ symbols. How can I find conjugacy classes of elements of $V$ using GAP commands? ( Since GAP has most commands for $p$-groups over finite field $F_p$ )
Conjugacy classes in GAP
gap
Best Answer
There probably is a better way of finding units -- what I will use is to work in the regular matrix representation, since it will give the units as a group of matrices.
Start with the regular matrix representation:
For performance reasons it turns out to be useful to change the basis so that the algebra is a direct sum of smaller dimensional algebras:
(Verify block form with
Display(mats[1]);
). Now we form the algebra:Now we start searching for units:
This runs a bit and gives us a group of matrices:
For efficiency we cnonvert the unit group (which happens to be solvable) to a
PcGroup
:and compute its conjugacy classes: