Prove that every automorphism of Z12 sends 6 to 6. Generalize this result to Z2n Prove...
2 6. Let n E N and z E C with |c| 1 and z2nメ-1. Prove that 122n 2 6. Let n E N and z E C with |c| 1 and z2nメ-1. Prove that 122n
33. Prove that 11n - 6 is divisible by 5 for every positive integer n.
21. Prove or disprove: Every nonabelian group of order divisible by 6 contains a subgroup of order 6.
(2) (4 pts) We are going to generalize the result from the previous exercise as follows. Fix two positive numbers c and c2 satisfying cI <c. Define number of primes between ciN and c2N number of integers between ciN and cN This is the probability that an integer n in the interval cN,cN s a prime number. Use the Prime Number Theorem to find an easy to compute function F(c,c2; N) such that P(C1,C2; N) lim (3) (3 pts each)...
(6)(20 points) (a) Let G be a cyclic group of order n. Prove that for every divisor dofn there is a subgroup of Ghaving order d. (b) Characterize all factor groups of Z70.
(6)(20 points) (a) Let G be a cyclic group of order n. Prove that for every divisor d of n there is a subgroup of G having order d. (b) Characterize all factor groups of Z70 -
(6) Prove that if H is a subgroup of Z, then there is a unique nonnegative integer m such that H = mZ. (7) Prove that every strictly increasing sequence of subgroups of Z is finite.
Problem 5. Let V and W be vector spaces, and suppose that B (vi, ..., Vn) is a basis of V a) Prove that for every function f : B → W, there exists a linear transformation T: V → W such that T(v;)-f(7) for all vEB (b) Prove that for any two linear transformations S : V → W and T : V → W, if S(6) = T(6) for all ï, B, then S = T (c) Prove...
Use mathematical induction to prove that the statement is true for every positive integer n. 1'3+ 24 +3'5 +...+() = (n (n+1)(2n+7))/6 a. Define the last term denoted by t) in left hand side equation. (5 pts) b. Define and prove basis step. 3 pts c. Define inductive hypothesis (2 pts) d. Show inductive proof for pik 1) (10 pts)
6. Prove that a tangent line to a circle is perpendicular to a radius of that circle at the point of tangency. (This is the converse of the result proved by contradiction on page 93.)