Examples of groups where any non-identity element has order $n$.

Question

Is the following proposition true?

For every $n \in \mathbb N$ there exists a group $G$ such that every element $g \in G\setminus \{e\}$ has order $ord(g)=n$.

Before someone asks, this isn't homework - curiosity rather.

Answer 1

No. With the exception of the trivial group $\{e\}$, for which the condition is vacuously true regardless of $n$. And so I will assume you meant "(...) there exists a non-trivial group $G$ (...)".

If $p$ is a prime number dividing $n$ and $g$ is of order $n$, then $g^{n/p}$ is of order $p$. Such $n$ will be a counterexample as long as $n\neq p$. And so this excludes all composite $n$.

On the other hand "yes" if $n=p$ is prime. One such example is $(\mathbb{Z}_p)^k$ for any cardinal (even infinite) $k\geq 1$. For an interesting infinite, nonabelian group such that every nontrivial element is of order $p$ see Tarski monster group.