Abstract Algebra Answers

For G = S3 and any field k of characteristic 2, view V = ke1 ⊕ ke2 ⊕ ke3/k(e1 + e2 + e3) as a (simple) kG-module with the permutation action. Show that kG ∼ M2(k)×(k[t]/(t2)), and that kG/rad(kG) ∼ M2(k)×k.

For G = S3 and any field k of characteristic 2, view V = ke1 ⊕ ke2 ⊕ ke3/k(e1 + e2 + e3) as a (simple) kG-module with the permutation action. Show that W = V ⊗k V with the diagonal G-action is not a semisimple kG-module.

Let U, V be simple modules of infinite k-dimensions over a group algebra kG (where k is a field), and let W = U ⊗k V , viewed as a kG-module under the diagonal G-action. Does W have finite length?

Let G be the group of order 21 generated by two elements a, b with the relations a^7 = 1, b^3 = 1, and bab^−1 = a^2. How about RG and the real representations of G?

Let G be the group of order 21 generated by two elements a, b with the relations a^7 = 1, b^3 = 1, and bab^−1 = a^2. Construct the irreducible rational representations of G and determine the Wedderburn decomposition of the group algebra QG.

Let G be the group of order 21 generated by two elements a, b with the relations a^7 = 1, b^3 = 1, and bab^−1 = a^2. Construct the irreducible complex representations of G, and compute its character table.

Let G be a cyclic group of prime order p > 2. Show that the group of units of finite order in QG decomposes into a direct product of G with {±1} and another cyclic group of order 2.

Let G = S3, and k be any field of characteristic 3. It is known that there are exactly six (finite-dimensional) indecomposable representations for G over k. Construct these representations.

Let G = S3, and k be any field of characteristic 3. Show that there are only two irreducible representations for G over k, namely, the trivial representation and the sign representation.

Let e = (Sum over g∈G) a_g * g ∈ kG be an idempotent, where k is a field and G is a finite group. Let χ be the character of G afforded by the kG-module kG • e. Show that for any h ∈ G, χ(h) = |CG(h)| (Sum over g∈C)•a_g,
where C denotes the conjugacy class of h^−1 in G.