Abstract Algebra Answers

If G is an f. c. group and is finitely generated, show that [G,G] is finite.

Let G be an f. c. group. Show that [G,G] is torsion.

Let G be a group such that [G : Z(G)] < ∞. Show that the commutator subgroup [G,G] is finite.

Let G be a group generated by x1, . . . , xn where each xi has finite order and has only finitely many conjugates in G. Show that G is a finite group.

Assume char k = 2, and let G = A • <x>, where A is the infinite cyclic group <y>. Show that R = kG is J-semisimple (even though G has an element of order 2).

figure out the following problem
10 divided 5 x 3 x 3....using order or operations

Assume char k = 2. Let A be an abelian 2'-group and let G be the semidirect product of A and a cyclic group <x> of order 2, where x acts on A by a → a^−1. If A is infinite, show that kG has no nonzero nil ideals.

Assume char k = 2. Let A be an abelian 2'-group and let G be the semidirect product of A and a cyclic group <x> of order 2, where x acts on A by a → a^−1. If |A| < ∞, show that rad kG = k (Sum over g∈G)•g, and (rad kG)^2 = 0.

Assume char(k) = 3, and let G = S3. Determine the index of nilpotency for J, and find a k-basis for J^i for each i.