
group theory

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group theory QoDetermine thegrup GKablab, (ab)*>? Consudes the preodatin asblå,B,ab}7 Deteraine the elements...
1. Let a and b be elements of a group
. Prove that ab and ba have the same order.
2. Show by example that the product of elements of nite order in a
group need not
have nite order. What if the group is abelian?
group theory
2. Consider the group presentation (a,b a1,b3,aba b Determine a van Kampen lemma word for W ab ab dint Inr a. 0
2. Consider the group presentation (a,b a1,b3,aba b Determine a van Kampen lemma word for W ab ab dint Inr a. 0
Let a and b be elements of a group G such that b has order 2 and
ab=ba^-1
12. Let a and b be elements of a group G such that b has order 2 and ab = ba-1. (a) Show that a” b = ba-n for all integers n. Hint: Evaluate the product (bab)(bab) in two different ways to show that ba+b = a-2, and then extend this method. (b) Show that the set S = {a”, ba" |...
= 12. List the elements and draw a multiplication table of the group (a,b a' = 1, b 1. (ab) = 1); prove that you have the required group. Do you recognize this group?
2. Let G be an abelian group. Suppose that a and b are elements of G of finite order and that the order of a is relatively prime to the order of b. Prove that <a>n<b>= <1> and <a, b> = <ab> .
1. The equilibrium constant for the decomposition AB into its elements A and B has an equilibrium constant of 0.0900 at a given temperature. What is the equilibrium concentration of AB if 0.050 M AB is allowed to decompose? 2 AB (g) = Az (g) + B2 (g)
Let D4 be dihedral group order 8. So D4={e, a, a^2, a^3, b, ab, a^2b, a^3b}, a^4 = e, b^2= e, ab=ba^3; A. FIND ALL THE COSETS OF THE SUBGROUP H= , list their elements. B. What is the index [D4 : H] C. DETERMINE IF H IS NORMAL
1. Suppose a and b are elements of a group G. Prove, by induction, (bab−1)n = banb−1 . Hence prove that if a has order m, then bab−1 also has order m. Deduce from question (#1) that in any group ab and ba have the same order (you may assume ab has finite order). The assertion in Question (#1) can be generalized to an assertion about isomorphisms. State and prove it.
7. In the following cases, prodict ABO phenotypes and possible genottypes of offispring Dad Group AB Dad Group O Dad Group O a Mom Group A b. Mom Group B c. Mom Group A:B
Linear algebra and matrix theory: Show that if matrices A and B are such that AB = BA, then A and B have at least one common eigenvector.