Abstract Algebra Answers

8x^5+10x^4=4x^3+5x^2

2n^4-9n^2+=0

Let k be any field of characteristic 3, G = S3 and let V be the kG-module ke1 ⊕ ke2 ⊕ ke3/k(e1 + e2 + e3),
on which G acts by permuting the ei’s. Determine Spank(G) and its Jacobson radical.

Let k be any field of characteristic 3, G = S3 and let V be the kG-module ke1 ⊕ ke2 ⊕ ke3/k(e1 + e2 + e3),
on which G acts by permuting the ei’s. What is its unipotent radical?

Let k be any field of characteristic 3, G = S3 and let V be the kG-module ke1 ⊕ ke2 ⊕ ke3/k(e1 + e2 + e3),
on which G acts by permuting the ei’s. Is G a completely reducible linear group?

Let k be any field of characteristic 3, G = S3 and let V be the kG-module ke1 ⊕ ke2 ⊕ ke3/k(e1 + e2 + e3),
on which G acts by permuting the ei’s. Show that this realizes G as a linear group in GL(V ).

Can every finite group be realized as an irreducible linear group?

Let G ⊆ GLn(k) be a linear group over a field k. Show that every finite group can be realized as a linear group, but not every infinite group can be realized as a linear group.

For any field k of characteristic p, let G = SL2(Fp) act on the polynomial ring A = k[x, y] by linear changes of the variables {x, y}, and let Vd ⊆ A (d ≥ 0) be the kG-submodule of homogeneous polynomials of degree d in A. It is known (and thus you may assume) that V0, . . . , Vp−1 are a complete set of simple modules over kG.
If {d1, . . . , dn} is any partition of p, show that the tensor product Vd1 ⊗k • • •⊗k Vdn (under the diagonal G-action) is not semisimple over kG, unless p = 2and n = 1.