Abstract Algebra Answers

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. Is W semisimple?

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.

Suppose the character table of a finite group G has the following
two rows:
g1 g2 g3 g4 g5 g6 g7
μ 1 1 1 ω2 ω ω2 ω
ν 2 −2 0 −1 −1 1 1
where ω = e^2πi/3. Determine the rest of the character table.