# Does there exist a $n \in \mathbb{N},\; n>1$ such that the number of groups of order $n$ equals $n$?

Does there exist a $$n \in \mathbb{N},\; n>1$$ such that the number of groups of order $$n$$ equals $$n$$?

Me and my friends were just curious and checked the first 1000 numbers (using the internet).

– lhf
Mar 9, 2020 at 16:15
• I check up to order $2047$ and the answer is no. A somewhat close hit above $1000$ is $f(1056)=1028$. Mar 9, 2020 at 16:17
• This is open, and has been discussed many times before here, see for example Is the guess $moa(31)=11774$ in the moa function the true value? Mar 9, 2020 at 18:29
• I don't understand the votes to close. This is a legitimate question (and there isn't much the OP could have tried on their own). Mar 9, 2020 at 18:44
• In a paper about gnus, moas and other exotica , the author mentioned the conjecture that there is probably no such $n$ , but the question is , as mentioned , open. The answer is "no" for squarefree numbers , cubefree numbers upto $50\ 000$ , prime powers with exponent not exceeding $4$ and numbers with at most $3$ prime factors (multiplicity considered). This list is not complete, but shows some classes which can be ruled out. May 11, 2020 at 9:59

Let $$a(n)$$ be number of groups of order $$n$$. There is a conjecture from Conway, Dietrich and O'Brien (see also O'Brien's website here) that the sequence $$n, a(n), a(a(n)), a(a(a(n))), \dots$$ eventually consists of an infinite sequence of $$1$$.
The number $$n$$ you are looking for would be a counterexample to this conjecture, which is currently open (and fully checked up to $$2047$$).
• To be clear: unless I am not understanding, it's perfectly possible that an elementary argument would show that there was no $n$ of the desired form. I agree that your argument suggests that we will fail to find an example, but a negative solution should still be possible, yes?
• Sure, the non-existence of a fixed point $n$ would not imply that the conjecture is true (you could have cycles or you could have unbounded growth) Mar 9, 2020 at 16:55