Let $A$ be a local(not necessarily commutative) finite dimensional algebra over a (finite if it helps) field $K$. Is there a way to check with GAP (without using QPA as we do not know quiver and relations for A in general) to quickly check whether $A$ is a Frobenius algebra? Note that this is equivalent to the condition that the vector space dimension of the socle of the regular module $A$ is one-dimensional.
A concrete situation I have in mind is this: Let $B$ be the algebra $K[x_1,...,x_n]/(x^i_^2)$ (which is the group algebra of the elementary abelian 2-groups when the characteristic of $K$ is 2) and $A$ a subalgebra with identity generated by some elements of $B$ of positive degree. $B$ could also be another group algebra of a $p$-group such as the quaternions.
Here is a concrete example (constructes using QPA, so you have to do LoadPackage("qpa") first in GAP):
Q:=Quiver(1,[[1,1,"x"],[1,1,"y"],[1,1,"z"]]);KQ:=PathAlgebra(GF(3),Q);AssignGeneratorVariables(KQ);rel:=[x*y-y*x,y*z-z*y,x*z-z*x,x^2,y^2,z^2];B:=KQ/rel;Dimension(B); gens:=GeneratorsOfAlgebra(B);e:=gens;x:=gens;y:=gens;z:=gens; A:=Subalgebra(B,[e,x+y+z,x*y,x*y*z]);O:=Basis(A);Dimension(A);