Vtyjkyuk

Define a nearly immaculate group as a finite group $G$, such, that the sum of the orders of all its normal subgroups is $2|G| - 1$. It is quite obvious to see, that all groups of the form $C_2^n$ are nearly immaculate. But are there any not of the type?

One can see, that a cyclic group $C_n$ is nearly immaculate iff $n$ is almost perfect. Whether all almost perfect numbers are of the form $2^n$ or not, is an open problem. But, what’s about non-cyclic nearly immaculate groups? Do they exist?

I do not have a full answer, but I can show that such an example cannot be a $p$-group. It seems challanging to generalize this to nilpotent groups.

Claim: among the (finite) $p$-groups, $Gcong C_2^n$ are the only examples.

Proof: If $pneq 2$, then the order of every nontrivial normal subgroup is divisible by $p$, so the sum of orders is congruent to $1$ modulo $p$.
Hence, it cannot be $2|G|-1$.
If $p=2$ and $G=2^n$, then there exists a normal subgroup in $G$ of order $2^k$ for every $0leq kleq n$.
As the sum of these orders is $2^n+1-1= 2|G|-1$, there must be exactly one of each.
In particular, there is a unique maximal normal subgroup $Mtriangleleft G$, and thus any element $gnotin M$ generates $G$.