Abstract Algebra Answers

Let k be an algebraically closed field, and G ⊆ GLn(k) be a completely reducible linear group. Show that G is abelian iff G is conjugate to a subgroup of the group of diagonal matrices in GLn(k).

Let G be a maximal unipotent subgroup of GLn(k) over a field k. If char(k) = p > 0. and n ≤ p, show that G has exponent p, and that it is the union of a family of subgroups Hi ∼ k such that Hi ∩ Hj = {1} for all i <> j.

Let G be a maximal unipotent subgroup of GLn(k) over a field k. If char(k) = p > 0, show that G is uniquely r-divisible for any positive integer r that is prime to p.

Let G be a maximal unipotent subgroup of GLn(k) over a field k. If char(k) = p > 0, show that G is a p-group.

Let G be a maximal unipotent subgroup of GLn(k) over a field k. If char(k) = 0, show that G is a torsionfree, nilpotent, and uniquely divisible group, and that it is the union of a family of subgroups Hi ∼ k such that Hi ∩ Hj = {1} for all i <> 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). If J is commutative under multiplication, show that G ∼ J as (abelian) groups.

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 subgroups y^k and z^k of G are either the same or have trivial intersection, and G is the union of all such subgroups.