Cyclic Groups: Definitions              QUIZ 3.1

1. Element g is a generator of group G if every element of G occurs as a ‘power’ of g. (a) T
(b) F
     
2. (Z, +) is a cyclic group having two generators. (a) T
(b) F
     
3. Every element of a group G generates a cyclic subgroup of G. (a) T
(b) F
     
4. The group {(1), (1 2 3), (1 3 2)} is a cyclic group of order 3. (a) T
(b) F
     
5.  The set {1, 3, 7, 9} under multiplication modulo 10 forms a cyclic group of order 4. (a) T
(b) F
     
 6. A cyclic group of even order has exactly one element of order 2. (a) T
(b) F
Check answers
1. (a) This is what we mean by G = <g>.


2. (a) (Z, +) is generated by 1 and –1.

3. (a) Taking the ‘powers’ of g either generates G itself, (in which case G is cyclic), or a cyclic subgroup.

4. (a) The group has 3 elements and so has order 3. It is generated by (1 2 3) or by (1 3 2).

5. (a) This cyclic group of 4 elements is generated by each of 3, 7 and 9.

6. (a) If G has order 2n, and is generated by g, then the nth power of g is the only element of order 2.