Smallest subgroup of a group that contains an element x

Question

I came across the following question on a YouTube video. It asked the following question: if we take the set of all integers under addition, what is the smallest group that contained the element x?

They answered it like so. To have a group, we would need $(x,0,-x)$ since x needs to have an identity, and an inverse. But then they said that we also needed all powers of x, so that the group satisfies the properties of closure under addition? I am not totally sure what "powers" in this context means, and why we need it to maintain closure for this subgroup. We can add any two numbers in this subgroup and still get an element in this subset. So why do we need all "powers" of x?

The "powers" of $x$ are not powers, because the group composition is addition. — Dietrich Burde, Sep 17 '21 at 12:26

score 1 · Accepted Answer · answered Sep 17 '21 at 12:29

1

When working with groups "powers" usually means with respect to the operation. In this case, we are dealing with addition.

Therefore when working with the group $G(\mathbb{Z},+)$ we have that $x^2 = x + x$. They are making sure that the additivity axiom is satisfied that is $\forall x,y \in G, x+y \in G$.

answered Sep 17 '21 at 12:29

Igor

549

But in this case, the only possible combinations are x-x, and x+0, and x-0, right? And these are all in the subset? – Sep 17 '21 at 12:30
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You have $x + x$ as well! – Igor Sep 17 '21 at 12:34
Oh yes! Thanks! – Sep 17 '21 at 12:36

Smallest subgroup of a group that contains an element x

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