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Clearly, such $d$ would divide the group order, and it is divided by the order of every single element. I wonder if there is a name for such integer, and if there are certain basic properties beyond the ones I just mentioned.

Shaun
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Daniel
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    It's called the exponent of the group. See https://math.stackexchange.com/questions/1228513/what-is-the-exponent-of-a-group – rogerl Sep 23 '21 at 17:19
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    In general, we say $G$ is "of exponent $n$" if $x^n=e$ for all $x\in G$. While some people use "exponent" to describe the least such $n$, this use is not universal. – Arturo Magidin Sep 23 '21 at 17:20
  • Thank you, @rogerl. Right to the point. I would encourage you to submit this as an answer so that I can close the question. – Daniel Sep 23 '21 at 17:23
  • Also, thanks @ArturoMagidin for the additional note. – Daniel Sep 23 '21 at 17:23

2 Answers2

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This is usually called the exponent of the group. See for example here or here.

rogerl
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It is known as the exponent of the group. The group need not be finite, either; for instance: $\prod_{n=0}^\infty \Bbb Z_2$ has exponent two.

Shaun
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