Normal Subgroups

Please see the attached file for the fully formatted problem. Let G be a group and let D ={(a,a,a):a E G}. Prove that D is a normal subgroup of G+G+G if and only if G is Abelian.

Abelian Groups

Find examples of the following. Explain your answers. (a) A nonabelian group G and a proper normal subgroup S such that G/S is cyclic.

Kernel of Homomorphism

Please see the attached file for the fully formatted problems. Suppose  is an onto homomorphism from ℤ16 to a group G of order 4. Find ker(). Explain your answer.

Element of Finite Order

Prove that every element of Q/Z has finite order, where Q is the set of rational numbers with group operation + and Z is the set of integers.

Exponential Function: Radioactivity

Complete the table for the given radioactive isotope. Isotope 226^Ra Half-life(years) 1620 Initial Quantity ______ Amount after 1000 yrs. 1.5g

Exponential and Logarithmic Equations : Modeling

The number of trees per acre N of a certain species is approximated by the model N=68(10^-0.04x), 5 <_ x <_ 40 (<_ = less than or = to) Where x is the average diameter of the trees (in inches) three feet above the ground. Use the model to approximate the average diameter of the trees in a test plot when N=21.

Exponential and Logarithmic Equations

The population P of a city is P=240,360e^0.012t where t=0 represents 2000. According to this model, when will the population reach 275,000?

Exponential and Logarithmic Functions

Radioactive Decay: Carbon 14 dating assumes that the carbon dioxide on earth today has the same radioactive content as it did centuries ago. If this is true, the amount of carbon 14 absorbed by a tree that grew several centuries ago should be the same as the amount of carbon 14 absorbed by a tree growing today. A peice of anc ...continues

(a) Complete the model by solving for K. (b) Estimate the number of units that will be sold if advertising expenditures are raised to $700.

Sales and Advertising: The sales S (in thousands of units) of a product after x hundred dollars is spent on advertising is S=10(1-e^kx). When $500 is spent on advertising, 2500 units are sold. (a) Complete the model by solving for K. (b) Estimate the number of units that will be sold if advertising expenditures are raised t ...continues

Exponential and Logarithmic Functions

Endangered Species: A conservation organization releases 100 animals of an endangered species into a game preserve. The organization believes that the preserve has a carrying capacity of 1000 animals and that the growth of the herd will follow the logistic curve. p(t)=1000/1+9e^-0.1656t where t is measured in months (a) ...continues