For a finite group G, let |G| denote the order of G and write $D(G) = \sum_{N \triangleleft G} |N|$, the sum of the orders of the normal subgroups.  I would like to call G "perfect" if D(G) = 2|G|, since then the cyclic group of order n is perfect if and only if the number n is perfect.  But the term "perfect group" is taken, so I'll call such a group immaculate.

My question is:


  Does there exist an immaculate group of odd order?


Since the cyclic immaculate groups correspond one-to-one with the perfect numbers, a "no" answer would immediately prove the famous conjecture that there are no odd perfect numbers.  However, perhaps someone can easily see that there is a non-cyclic immaculate group of odd order, proving that the answer is "yes".

Here's what I know.  There are no abelian immaculate groups except for the cyclic ones.  (Edit: more generally, if $D(G) \leq 2|G|$ then every abelian quotient of $G$ is cyclic.  Proof: not hard, and given here.)  However, there do exist nonabelian immaculate groups, e.g. $S_3 \times C_5$ (of order 30).  Derek Holt has computed all the immaculate groups of order less than or equal to 500.  Their orders are
$$
6, 12, 28, 30, 56, 360, 364, 380, 496
$$
(Integer Sequence A086792).  Of these, only 6, 28 and 496 are perfect numbers; the rest correspond to nonabelian immaculate groups.  Some nonabelian immaculate groups of larger order are also known, e.g. $A_5 \times C_{15128}$, $A_6 \times C_{366776}$, and, for each even perfect number n, a certain group of order 2n.  But these, too, all have even order.

Edit: Steve D points out that p-groups can never be immaculate.  This also appears as Example 2.3 here; it follows immediately from Lagrange's Theorem.  I should have mentioned this, as it rules out an easy route to a "yes" answer.

I did a little computer search and I think I found an example of an odd immaculate group.

I searched for groups of the form $G=(C_q \rtimes C_p) \times C_N$ with odd primes $p,q$ such that $p | q-1$ and $N$ an odd integer satisfying $(N,pq)=1$. Using Tom's notations and results, we have

\begin{equation*}
\frac{D(G)}{|G|} = \frac{D(C_q \rtimes C_p)}{|C_q \rtimes C_p|} \cdot \frac{D(C_N)}{|C_N|} = \frac{1+q+pq}{pq} \cdot \frac{\sigma(N)}{N}
\end{equation*}
where $\sigma(N)$ denotes the sum of divisors of $N$. We want $\frac{\sigma(N)}{N} = \frac{2pq}{1+q+pq}$. Since the last fraction is irreducible, $N$ has to be of the form $N=(1+q+pq)m$ with $m$ odd. I found the following solution :

\begin{equation*}
p=7, \quad q=127, \quad m=393129.
\end{equation*}
This gives the immaculate group $G=(C_{127} \rtimes C_7) \times C_{399812193}$, which has order $|G| = 355433039577 = 3^4 \cdot 7 \cdot 11^2 \cdot 19^2 \cdot 113 \cdot 127$.

Edit : here is the beginning of an explanation of "why" $m$ is square in this example ($393129=627^2$). Recall that an integer $n \geq 1$ is a square if and only if its number of divisors is odd (consider the involution $d \mapsto \frac{n}{d}$ on the set of divisors of $n$). If $n$ is odd, then all its divisors are odd, so that $n$ is a square if and only if $\sigma(n)$ is odd. Now consider $N=(1+q+pq)m$ as above. The condition on $\sigma(N)/N$ implies that $\sigma(N)$ is even but not divisible by $4$.

If we assume that $1+q+pq$ and $m$ are coprime, then $\sigma(N)=\sigma(1+q+pq) \sigma(m)$, so the reasoning above shows that $1+q+pq$ or $m$ is a square (but not both). If $1+q+pq=\alpha^2$ then $\alpha \equiv \pm 1 \pmod{q}$ so that $\alpha \geq q-1$, which leads to a contradiction. Thus $m$ is a square (it is possible to show further that $1+q+pq$ is a prime times a square).

If $1+q+pq$ and $m$ are not coprime, the situation is more intricate (this is what happens in the example I found : we had $\operatorname{gcd}(1+q+pq,m)=9$). Let $m'$ be the largest divisor of $m$ which is relatively prime to $1+q+pq$. Put $m=\lambda m'$. Then $\lambda(1+q+pq)$ or $m$ is a square. I don't see an argument for excluding the first possibility, but at least if $\lambda$ is a square then so is $m$.
