2. Suppose that < a> is a cyclic group of order 10. Find all the generators...
Find all the generators of each of the following cyclic groups a. The even numbers under addition b. c. d. {Z20, +} <a> where a = 15 <a> where al = 15
(a) List all the generators of < 5 > in Z6o. (b) List all of the left cosets of < 10 > in the subgroup < 2 > of Z60.
4. Suppose G is a group of order n < 0. Show that if G contains a group element of order n, then G is cyclic.
(a) List all the generators of < 5 > in Z60. (b) List all of the left cosets of < 10 > in the subgroup < 2 > of Z60. 7. (7 points each) (a) List all the generators of <5> in Z6o. (b) List all of the left cosets of <10 > in the subgroup < 2 > of Z60-
Question 3. Use the class equation to show that, if G is a group of order p' for p a prime and r > 1 (such a G is called a p-group), then Z(G) must be nontrivial
Please show all your work for this problem. Thank you so much in advance. (If you do this using left hand and right hand limits, please show every single step). 3) Let f(x) = x for x < 1, and ax2 + b for 2 > 1. Find all (a, b) such that f(x) is differentiable on all of R.
Prove using mathematical induction that 3" + 4" < 5" for all n > 2.
4. (10 points) Find the solution to the wave problem Ut = c+421 +COSI, <0, t>0, with initial conditions u(1,0) = sin r, 4(1,0) = 1+I.
Suppose that the cdf of random variable X is Skex-3, X <4 11, X >4 Find k. (2 Points) Find the expected value of X. (2 Points)
5. Prove that U(2") (n > 3) is not cyclic.