For the group G = <Z19*, × > a. Find the order of the group. b. Find the order of each element in the group. c. Find the number of primitive roots in the group. d. Find the primitive roots in the group. e. Show that the group is cyclic.

a. The order of the group G = <Z19*, × > is the number of elements in the group. Since Z19* is the set of integers less than 19 and relatively prime to 19, the order of the group is φ(19) = 18, where φ is the Euler's totient function.

b. The order of an element a in a group is the smallest positive integer n such that a^n = e (the identity element). In this case, the identity element is 1. So, we need to find the smallest n such that a^n = 1 mod 19 for each a in Z19*. This can be done by trial and error or by using the Euler's theorem.

c. The number of primitive roots in a group is given by φ(φ(n)) where φ is the Euler's totient function. So, the number of primitive roots in this group is φ(φ(19)) = φ(18) = 6.

d. The primitive roots of a group are the elements whose order is equal to the order of the group. In this case, the primitive roots are the elements a in Z19* such that a^18 = 1 mod 19 but a^n ≠ 1 mod 19 for any n < 18. This can be found by trial and error.

e. A group is cyclic if there exists an element a in the group such that every element in the group can be written as a power of a. In this case, since there are primitive roots, the group is cyclic. Specifically, if g is a primitive root, then every element in the group can be written as g^n for some integer n, which shows that the group is cyclic.