Abstract Algebra Answers

Let J be a nil ideal in an algebra R over a field k of characteristic 0, and let G be the group 1 + J ⊆ U(R). For any y ∈ G, let y^k := {y^α : α ∈ k}. If y <> 1, show that y^k is a subgroup of G isomorphic to the additive group k.

Let J be a nil ideal in an algebra R over a field k of characteristic 0, and let G be the group 1 + J ⊆ U(R). For any y ∈ G, let y^k := {y^α : α ∈ k}. If y <> 1, show that y^k is a subgroup of G isomorphic to the additive group k.

Let J be a nil ideal in an algebra R over a field k of characteristic 0, and let G be the group 1 + J ⊆ U(R). Show that group G is torsionfree.

Let J be a nil ideal in an algebra R over a field k of characteristic 0, and let G be the group 1 + J ⊆ U(R). Show that G is a uniquely divisible group; that is, for any positive integer r, each element of G has a unique rth root in G.

Let J be a nil ideal in an algebra R over a field k of characteristic 0, and let G be the group 1 + J ⊆ U(R). For any y ∈ G and α ∈ k, define y^α = exp(α log(y)) ∈ G. Show that log(yα) = α log(y), (y^α)^β = y^αβ, and y^α*y^β = y^(α+β) for any α, β ∈ k.

Let J be a nil ideal in an algebra R over a field k of characteristic 0, and let G be the group 1 + J ⊆ U(R). Similarly, show that two elements y, y' ∈ G commute iff log(y) and log(y') commute in J, in which case we have log(yy') = log(y) + log(y') ∈ J.

Let J be a nil ideal in an algebra R over a field k of characteristic 0, and let G be the group 1 + J ⊆ U(R). Show that two elements x, x' ∈ J commute iff exp(x) and exp(x') commute in G, in which case we have exp(x) • exp(x') = exp(x + x') ∈ G.

Let J be a nil ideal in an algebra R over a field k of characteristic 0, and let G be the group 1 + J ⊆ U(R). Show that the map exp : J → G defined by the Taylor series of the exponential function is a one-one correspondence between J and G, with inverse given by the log-function.

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.