It is known from Serre's classical result that every p-torsion occurs in the homotopy groups of every sphere. Is it known: do elements of every order occur in homotopy groups of spheres?



Do elements of every order occur in homotopy groups of spheres?
As others suggested, I am posting my earlier [comment](https://mathoverflow.net/questions/427273/do-elements-of-every-order-occur-in-homotopy-groups-of-spheres#comment1098754_427273) as an answer:




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$\DeclareMathOperator\denom{denom}$Sure. Let $n$ be a positive integer. Let $N$ be the product of $2n$ and Euler $\phi(2n)$. Then for each prime $p$ that divides $n$, the number $p-1$ divides $N$. So for each prime divisor $p$ of $n$, the denominator of the Bernoulli number $B\_N$ is divisible by $p$. So none of the prime factors of $n$ cancel with factors in the numerator of $B\_N/N$. So $\denom(B\_N/N)$ is divisible by $n$. Now $\denom(B\_N/N)$ or $2\denom(B\_N/N)$ is the order of a cyclic summand in the image of the J-homomorphism in the $(2N-1)$st stable homotopy group of $S^0$. So by Freudenthal, there is an unstable homotopy group of a sphere with an element of order $n$.


