[Math] Generators of the additive integer group


My question is a simple one, perhaps it is more of a notational question but I am not sure.

In Dummit + Foote's Abstract Algebra, when introducing generators (page 26 section 1.2 Dihedral groups), they state that:

For example, the integer 1 is a generator for the additive group $\mathbb{Z}$ of integers since every integer is a sum of a finite number of +1's and -1's, so $\mathbb{Z}=\langle 1\rangle$.

My question is that since every element of ($\mathbb{Z},+$) can be written with +1's and -1's, shouldn't the generator be $\langle 1,-1\rangle$? Or is it notation to not include the inverse elements.

Thank you in advance.

Best Answer

$\langle 1 \rangle$ is already a generator for $\mathbb{Z}$, because a generator also "generates" inverse elements. This is due to the definition of a generator.

From Wikipedia:

In abstract algebra, a generating set of a group is a subset such that every element of the group can be expressed as the combination (under the group operation) of finitely many elements of the subset and their inverses.

So $-3\in \langle 1 \rangle$, because we have $-3= (-1)+(-1)+(-1)$.

The same goes for $\langle -1\rangle$.

Also note that technically $\langle 1, -1\rangle$ is also a valid generator of the group, but it happens that there is also a "smaller" generator.

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