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I do not know much about group theory, so sorry in case this question is not for MO. For a finite group $G$, denote by $f(G)$ the number of elements of the automorphism group of $G$.

Question: For which $n$ is there a unique group with $n$ elements such that $f(G)$ is minimal among all groups with n elements? What is the sequence $a_n$ giving the minimal $f(G)$ for a given $n$.

For $n \leq 63$ (except $n=32$ which has too many groups for my computer it seems and I was not able to finish the calculation with GAP. Ill try it after the holidays with a better computer) it was true that there is a unique such group namely $G=Z/Zn$, so $a_n$ is the Euler totient function $\phi(n)$ for those values.

This suggests the following question:

Question: Do we have $f(G) \geq \phi(n)$ in case $G$ has $n$ elements?

Martin Sleziak
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Mare
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1 Answers1

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The following conjecture is problem 15.43 in The Kourovka Notebook, attributed to M. Deaconescu :

Let $G$ be a finite group of order $n$.

a) Is it true that $|Aut(G)|\geq \varphi (n)$ where $\varphi$ is the Euler function?

b) Is it true that $G$ is cyclic if $|Aut(G)|=\varphi (n)$?

The problem is now in the solved section. In fact J. N. Bray and R. A. Wilson in the paper "On The Orders of Automorphism Groups of Finite Groups" show that $\frac{|Aut(G)|}{\varphi(|G|)}$ can be arbitrarily close to zero. And the same holds if we add the extra restrictions of $G$ being perfect or soluble as shown in "On The Orders of Automorphism Groups of Finite Groups II"

The failure of the conjecture in this strong form (ratio can be arbitrarily close to zero) was shown to hold even when restricting to $p$-groups $G$ by J. González-Sánchez and A. Jaikin-Zapirain in "Finite p-groups with small automorphism group".

Community
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Gjergji Zaimi
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  • Thanks, is the smallest n known where a) fails? – Mare Dec 26 '19 at 17:06
  • @Mare I don't know if the exact value is known, but a) is true for all groups with $|Aut(G)|\le 48$. – Gjergji Zaimi Dec 26 '19 at 17:09
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    Maybe this is a good candidate for https://mathoverflow.net/questions/15444/examples-of-eventual-counterexamples . – Mare Dec 26 '19 at 17:11
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    A quasisimple counterexample to a) is a perfect group $G$ such that $G/Z(G)\cong PSL_3(4)$, of order $20160$, and $Z(G)\cong Z_2\times Z_4\times Z_3$. This may be the smallest quasisimple counterexample. It exists since the (unusually large) Schur multiplier of $PSL_3(4)$ is $Z_4\times Z_4\times Z_3$. – Richard Lyons Dec 26 '19 at 20:23