MAT 334 Quiz 2
February 11, 2005
1. Find an example of each of the following. For each of the following, the
answers may vary. But I will give a couple of examples for each one.
(a) An infinite group.
Zunder addition, Qunder addition.
(b) A group of order 5
Z5
(c) An Abelian group
Z,Z4,Z10, etc.
(d) A non-Abelian group
Dnfor any n > 3.
(e) A non-cylic group
Dnfor any n > 3.
2. Find the orders of all elements of Z10.
The orders of the elements 1,3,7 and 9 are all 10, since they all generate
Z10.
The orders of the elements 2,4,6, and 8 are all 5, since they all generate
{0,2,4,6,8}.
The order of the element 0 is 1, since it generates {0}.
The order of the element 5 is 2, since it generates {0,5}.
3. Find the orders of all elements of U(10).
The order of the element 1 is 1, since it generates {1}.
The order of the elements 3 and 7 is 4, since they generate {1,3,7,9}.
The order of the element 9 is 2, since it generates {1,9}.
4. List all the generators of the cyclic group Z20.
1,3,7,9,11,13,17 and 19
