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Let $n\in \Bbb N$. Any finite group $G$ with $|G|=n$ has a solution for the Diophantine equation $$\sum_{d|n}x_d\phi(d)=n~~~~~~~~~~~~~~~~~~~~~~~(1)$$ where $\phi$ is the Euler's totient function, $d$ is any positive divisor of $n$ and the $x_d$ are the unknowns (see this). Call such a solution a $G$-solution.

Are there number-theoretic conditions which characterize $G$-solutions of the equation where $G$ is a simple group?

Edit:

The comments say that I should add this:

Definition: Let $G$ be a group of order $n$. $(x_d)_{d|n}$ is a $G$-solution of equation $(1)$ if for each $d$, $x_d\phi(d)$ is the number of elements of $G$ with order $d$.

I'm looking for number-theoretic characterization of a {simple $G$}-solution, that is, a $G$-solution with $G$ a simple group.

It is clear that a $G$ produces a unique $G$-solution. I'm not completely sure, but I think a unique simple $G$ can produce a {simple $G$}-solution.

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    $\begingroup$ I do not understand how the fact that G is a simple group could matter: the G-solutions actually depend only on n. $\endgroup$
    – user40023
    Commented Aug 17, 2015 at 17:41
  • $\begingroup$ I'm asking about {simple $G$}-solutions. Not all $G$-solutions. $\endgroup$
    – Minimus Heximus
    Commented Aug 17, 2015 at 21:55
  • $\begingroup$ There should not be a meaningful characterization if $x_d$'s are not suitably defined. For example, let $x_d$ for $d>2$ be any number such that $\sum_{d|n}s_x\phi(d)\leq n-2$. Then we may choose $x_1$ and $x_2$ arbitrary in such a way that $\sum_{d|n}d_x\phi(d)=n$. So, two non-isomorphic groups can have many similar $G$-solutions. Indeed, the definition of $G$-solutions does not depends on the structure of the group $G$ at all. $\endgroup$
    – M. Farrokhi D. G.
    Commented Aug 24, 2015 at 10:27
  • $\begingroup$ I thought it was clear. $\endgroup$
    – Minimus Heximus
    Commented Aug 24, 2015 at 13:30

1 Answer 1

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It is indeed true that when the sequence $(x_d)_{d\mid n}$ comes from a finite simple group $G$, then $G$ is the only group (up to isomorphism) producing that sequence. In fact, $G$ is determined by its order $\lvert G \rvert$ and the set of its element orders, that is, the set of $d$'s such that $x_d \neq 0$. This has been proved using the classification of finite simple groups (Vasilʹev, A. V.; Grechkoseeva, M. A.; Mazurov, V. D.: Characterization of finite simple groups by spectrum and order, MR2640961 (2011b:20073)).

Since the finite simple groups are classified, you have in principle a parametrization of the possible "{simple $G$}-solutions", but I doubt if there is any useful number-theoretic characterization of these solutions, other than the parametrization itself. (In fact, I don't know whether there is any number-theoretic characterization of the orders of finite simple groups other than the list coming from the classification itself). Of course, there are various necessary conditions coming from various theorems in group theory. For example, when $(x_d)$ is a $G$-solution, then $(1/d)\sum_{e\mid d} x_e\phi(e)$ is an integer by a theorem of Frobenius, and by a conjecture of Frobenius which has been proved in the meantime (using CFSG), this integer is $\geq 2$ for all proper divisors $d$ of $\lvert G \rvert$ when $G$ is simple.

A famous problem of Thompson (still open, to the best of my knowledge) is whether the sequence $(x_d)$ (equivalently, the order type of a group) determines whether this group is solvable (Kourovka notebook, Problem 12.37). In general, it is a difficult problem to characterize solutions $(x_d)$ coming from (certain classes of) groups.

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