Question

Show that the only generators of the additive cyclic group areZ and 1 and -1 .

Short answer

Answer

It is proved that the only generators of the additive group are 1 and -1.

Step-by-step solution

Step by Step Solution: Step 1: Obtain the generators of Z

We know the definition of a cyclic group:

A group is called cyclic if there exists an element such that every element of can be written as aⁿ, for some $n\in Z$.

Here, a is called a generatorof the group.

$G=〈a〉,orG=\left(a\right)$

The generator of the group is denoted by:

Suppose is a generator of Z.

Then, $\exists b\in \mathbf{\text{Z}}$such that ab = 1.

This implies that a/1.

The only divisors are: 1 and -1.

Therefore, a generator can only be either 1 or -1.

Show the generators in the general case

Now, for every $n\in \mathbf{\text{Z}}$, we have:

n = n.1and n = (-n) . (-1)

Hence, 1 and generate Z.

Hence, the only generators of the additive cyclic group are 1 and -1.