Abstract Algebra Answers

A. A doctor writes a prescription for 20 mg of Tobramycin antibiotic saline solution. The solution is to be injected intravenously one time a day. On hand we have a 30 mg/ 2ml vial of Tobramycin, do the necessary calculations to give the patient the correct single dosage.

Prove that abelian group of order pq (p,q distinct prime) is cyclic.

Prove that if elements a,b of group G commutes, then LCM of their orders is multiply of order
of their product.

Prove that if all squares of all elements are equal identity of group, that it is abelian.

Suppose a nonhomogenous system of 9 linear equations and 10 variables has a solution for all possible values. Discuss whether it is possible for two nonzero solutions that are not multiples of each other to exist for the associated homogenous system.

Define transformation T: R^3 -> R^3 by T(x)=Ax where A (3x3) matrix and has eigenvalues 5 and -2. Is A similar to a diagonal matrix? explain

vicky is thinking a number a number less than 700.when she interchanges ones and hundreds digit, the number increases by 297. when she interchanges the ones and tens digit, the result increases by 18.can you find the number she thinking of?

A softdrink cup is cylindrical in shape. I f its volume is 1884cm3 and the radius of the circular base is 10cm, how deep is the cup?

If A, B are subgroups of G such that b^(-1) Ab⊇A for any b ϵ B, show that AB≤G.