Question

how many generators are there in an infinite cyclic group.
A.1
B.3
C.2
D. Infinite

Answer 1

Hint: Remember the definition of a cyclic group and take an example to obtain the required answer.

Complete step by step solution:
The definition of a cyclic group is: A group that contains a cyclic subgroup is known as cyclic subgroup.
The infinite cyclic group \[\left( {\mathbb{Z}, + } \right)\] has two generators 1 and -1.

Therefore, the correct option is C.

Additional information:
A cyclic group or monogenous group is a group that is generated by a single element in group theory, a branch of abstract algebra. An Abelian group is included in every cyclic group.A cyclic group has only cyclic subgroups.The order of each member in a finite cyclic group G that has order n divides n. If and only if an infinite group is finitely generated and has exactly two ends, then it is essentially cyclic. A group produced by a single element is known as a cyclic group. As a result, every other element of the group can be expressed as a power of an element called g, for example. The group's generator is this element g.

Note: Sometime students write the whole proof that an infinite cyclic group has two generators, but here the question is only to identify how many generators will be in a cyclic group the proof is not needed here. So, please go through the demand of the question and then answer the question properly.