Homework 2
数学assignment代做 Problem 1. Verify that each of the following sets is a group under the indicated binary operation. [You may assume in each case ···
Problem 1.
Verify that each of the following sets is a group under the indicated binary operation. [You may assume in each case that the operation is associative.] State whether each group is abelian.
Problem 2. 数学assignment代做
Let G be a fifinite abelian group, written multiplicatively.
a) Explain why we may unambiguously refer to the product of all the elements of G.
b) Let a be the product of all the elements in G. Show that a is equal to the product of all the involutions in G. [Hint. Involutions are self-inverse; elements of larger order are not.]
c) What is a²?
d) Suppose G is cyclic of order n with generator g. What is a? [Hint. Your answer should depend on the parity of n.]
Problem 3. 数学assignment代zuo做
Let G be a group, possibly infifinite. The exponent of G, denoted exp(G), is the least positive integer n such that gn = e for all g in G. If no such integer exists, we say G has infifinite exponent.
a) Give an example of a group with infifinite exponent but every element has fifinite order. [Hint. There’s one on this page.]
b) Suppose exp(G) is fifinite. Show that exp(G) = lcm{o(g) : g ∈ G}.
c) Let G be a fifinite group. Show that exp(G) | o(G)
d) Let G be a fifinite abelian group. Show that exp(G) = o(G) iffff G is cyclic. Give an example of a fifinite nonabelian group whose exponent and order are equal.
Problem 4. 数学assignment代做
Let H be the hyperbola xy = 1 and let O be the point (1, 1).
a) Find an explicit formula for P ⊕ Q in terms of the Cartesian coordinates of P and Q.
b) What is the ⊕-inverse of (x, y) in H ?
c) Find all elements of fifinite order in the group H. [Hint. For a) and b), imitate the worked example for the parabola group in Lecture 4.]
