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When trying to understand a notion, it often gives great insight to see it as the "shadow" of something bigger, carrying more information. The notion of categorification relies on this idea.


A basic example, where hopefully the point is clearly seen: if $n$ is a natural number and $\varphi$ denotes the Euler function, then the following identity holds: $$n=\sum_{d\mid n} \varphi(d)\tag{1}$$ Now if $\Phi_k(X)$ denotes the $k$-th cyclotomic polynomial, one can write $$X^n-1=\prod_{d\mid n}\Phi_d(X)\tag{2}$$ Then $(1)$ is a "shadow" of $(2)$, as it is obtained by taking the degree of the polynomials on both sides. Note that $(1)$ can be (and originally was) proved by much more elementary means that building the whole theory of cyclotomic polynomials, otherwise this observation would somewhat lose interest.


Definition: a perfect number is a natural number that is equal to the sum of all of its proper positive divisors.

Question: Is there something "above" this definition?

Attempt: A number-theorist's-perfect group is a finite group whose order is equal to the sum of the orders of all of its proper subgroups.

For instance, cyclic groups with perfect order satisfy the above definition, since a cyclic group has one subgroup for each factor of its order. What is less easy, and the point of my question, is whether this notion is in any way actually interesting.

Possibly more interesting if blanks can be filled: A nobody's-perfect group is a finite group satisfying [some conditions that could look interesting to a group-theorist and which imply that its order is equal to the sum of the orders of all of its proper subgroups].

In that last case one could start investigating the interest of that notion by trying to figure out whether there is a nobody's-perfect group of each perfect order.

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    $\begingroup$ Relevant: mathoverflow.net/questions/54851/… $\endgroup$ – Steve D Aug 7 '18 at 2:01
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    $\begingroup$ @SteveD Thanks. The idea of restricting oneself to normal subgroups makes great sense given that we're after a natural construction. Of course I'd take any ideas, not necessarily related to my initial thought. $\endgroup$ – Arnaud Mortier Aug 7 '18 at 13:43
  • $\begingroup$ (+1) for the illustration of a "shadow", never heard of it before! $\endgroup$ – Peter Aug 8 '18 at 23:37

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