Minimal order attaining function

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Definition

In words

The minimal order attaining function or moa function is the function $\mathrm {moa} :\mathbb {N} \to \mathbb {N}$ defined by $\mathrm {moa} (n)$ equal to the smallest number such that there are $n$ groups of that order up to isomorphism.

In terms of the group number function

We can define the minimal order attaining function in terms of the group number function, denoted $\mathrm {gnu}$ , which outputs the number of groups of a given order up to isomorphism:

$\mathrm {moa} (n)=\mathrm {min} \{m\in \mathbb {N} :\mathrm {gnu} (m)=n\}$ .

Question of well-definedness

It is not known whether $\mathrm {moa}$ is well-defined for all natural numbers. Equivalently, it is a conjecture as to whether or not the group number functions is surjective. In other words, it is possible that there is such a number for which there will never be that number of groups up to isomorphism of any given order.

Small values

$n$	$\mathrm {moa} (n)$	Relevant "groups of order" page
1	1	groups of order 1
2	4	groups of order 4
3	75	groups of order 75
4	28	groups of order 28
5	8	groups of order 8
6	42	groups of order 42
7	375	groups of order 375
8	510	groups of order 510
9	308	groups of order 308
10	90	groups of order 90

This function is not well understood, and some very small values of the function are not known. For example, it is conjectured that $\mathrm {moa} (33)=163293$ . This is the smallest such unknown value. As of May 2024, this has not been confirmed.

External links

The sequence Oeis:A046057 in the Online Encyclopedia of Integer Sequences gives the values of the minimal order attaining function.