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Abstract Algebra
Let G be a group such that [G : Z(G)] < ∞. Show that the commutator subgroup [G,G] is finite.
Abstract Algebra
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.
Abstract Algebra
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).
Abstract Algebra
Find a closed form expression for the energy E of the two-sided sequence x[n]= an , for 0 < a < 1.
Verify your result in Matlab by letting a = 0.5 and computing the partial energy sum for n
= −5:5 and n = −20:20. Please show your Matlab commands in your solution.
Verify your result in Matlab by letting a = 0.5 and computing the partial energy sum for n
= −5:5 and n = −20:20. Please show your Matlab commands in your solution.
Abstract Algebra
Give me stepwise solution how to solve for e.g 2 raised to 1/5 manully or using electronic calculator and without using log or scientific calculator
Abstract Algebra
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.
Abstract Algebra
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.
Abstract Algebra
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.
Abstract Algebra
Assume char(k) = 3, and let G = S3.Compute the Jacobson radical J = rad(kG), and the factor ring kG/J.
Abstract Algebra
Let k be a commutative ring and G be any group. If kG is left artinian, show that kG is right artinian.
