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Is it known a characterization of finite groups of order $n$ having exactly $n$ subgroups?

A supplementary question: are there abelian groups other than the trivial group and $\mathbb{Z}_2$ with this property?

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  • $\begingroup$ Any particular reason you want these two numbers to be the same? $\endgroup$
    – Qiaochu Yuan
    Commented Dec 4, 2015 at 3:39
  • $\begingroup$ Yes! I found only few finite groups with this property: the trivial group, $\mathbb{Z}_2$, $S_3$, ... and so on. $\endgroup$
    – Joachim Grieg
    Commented Dec 4, 2015 at 3:46
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    $\begingroup$ Reminds me of this recent question: mathoverflow.net/questions/224636/…. Remeber: two is a coincidence; it takes three to make a trend. $\endgroup$
    – Yoav Kallus
    Commented Dec 4, 2015 at 3:58
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    $\begingroup$ A dihedral group $D_{2n}$ has $\tau(n)+\sigma(n)$ subgroups. Finding those numbers $n$ satisfying the equation $\tau(n)+\sigma(n)=2n$ is itself a difficult problem. The values of $n\leq 10^7$ for which the above identity holds are $1$, $3$, $14$, $52$, $130$, $184$, $656$, $8648$, $12008$, $34688$ and $2118656$ among which $14$, $52$, $130$, $184$, $656$, $34688$ and $2118656$ are of the form $2^kp$ for some odd prime $p$. $\endgroup$
    – M. Farrokhi D. G.
    Commented Dec 4, 2015 at 4:33
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    $\begingroup$ @M.FarrokhiD.G., 130 is not of the form $2^kp$, $p$ prime. $\endgroup$
    – Gerry Myerson
    Commented Dec 4, 2015 at 4:35

1 Answer 1

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There are finite groups of order $n$ having exactly $n$ subgroups for $n = 1$, $2$, $6$, $8$, $28$, $36$, $40$, $40$, $48$, $54$, $72$, $\dots$, and this list is exhaustive for $n < 96$. The structures of the groups of order $< 96$ which satisfy the condition are as follows:

  • $1$,
  • ${\rm C}_2$,
  • ${\rm S}_3$,
  • ${\rm C}_4 \times {\rm C}_2$,
  • ${\rm D}_{28}$,
  • ${\rm C}_6 \times {\rm S}_3$,
  • $({\rm C}_{10} \times {\rm C}_2) \rtimes {\rm C}_2$,
  • ${\rm C}_2 \times ({\rm C}_5 \rtimes {\rm C}_4)$,
  • $({\rm C}_3 \times {\rm Q}_8) \rtimes {\rm C}_2$,
  • $(({\rm C}_3^2) \rtimes {\rm C}_3) \rtimes {\rm C}_2$, and
  • ${\rm C}_6 \times {\rm A}_4$.

The GAP SmallGroups Library Id numbers of these groups are as follows:

[ [ 1, 1 ], [ 2, 1 ], [ 6, 1 ], [ 8, 2 ], [ 28, 3 ], [ 36, 12 ], 
  [ 40, 8 ], [ 40, 12 ], [ 48, 17 ], [ 54, 5 ], [ 72, 47 ] ]

The existence of a general non-trivial "characterization" of such groups seems rather unlikely to me.

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