So I am trying to think of an example of a finite group of size $n$ for each $ n \gt 1 $, but nothing is coming to mind.

If it is a finite group denoted as $G$, then the order of G is is $|G|$, but I can't think of a group that satisfies this. I am just stuck and I am not sure if I am not understanding the question.

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    $\begingroup$ Every group satisfies this for some $n$. You just need, given some arbitrary $n$, to find a group that satisfies it for that particular $n$. There's one obvious example. For a hint, if $n$ is prime, there is exactly one group of that order: generalise those. $\endgroup$ – user3482749 Nov 30 '18 at 14:59
  • $\begingroup$ I don't see the obvious example. I am still trying to understand this. Is there anyway you can be more explicit. I am new to abstract algebra $\endgroup$ – Hawaiian Rolls Nov 30 '18 at 15:35
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    $\begingroup$ Wait, would $\mathbb{Z}_3$ work? Since $|\mathbb{Z}_n| = n$? $\endgroup$ – Hawaiian Rolls Nov 30 '18 at 15:38
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    $\begingroup$ Yes, $\mathbb{Z}_n$ will work for any $n$. It's a group, it has order $n$. $\endgroup$ – user3482749 Nov 30 '18 at 15:40

Here you go:

$$\mathbb{Z}/n \mathbb{Z}$$

  • $\begingroup$ Can you explain your answer. This is what I am getting from your answer. That if n = 2, then the size of G would be $\pm \frac{1}{2}$ which has two elements? I am new to abstract algebra so I apologize for not understanding this immediately. $\endgroup$ – Hawaiian Rolls Nov 30 '18 at 15:29
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    $\begingroup$ $\mathbb{Z}/n\mathbb{Z}$ is a common notation for what you're calling $\mathbb{Z}_n$. $\endgroup$ – user3482749 Nov 30 '18 at 15:40
  • $\begingroup$ I do not want to sound rude, bit would you please rethink your usage of the word "abstract algebra" in my opinion this is at most group theory 101, abstract algebra should in my opinion at least involve short exact sequences or morphismspaces $\endgroup$ – Enkidu Nov 30 '18 at 15:46
  • $\begingroup$ I used the word abstract algebra as that is the name of the course. The tag abstract-algebra references a huge list of topics, groups being one of them. I believe me using abstract algebra is not incorrect. @Enkidu $\endgroup$ – Hawaiian Rolls Nov 30 '18 at 15:49
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    $\begingroup$ Any course that involves the abstract theory of groups definitely deserves the name "abstract algebra". $\endgroup$ – Tobias Kildetoft Nov 30 '18 at 16:06

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