# Does the cyclic group $C_n$ have the fewest automorphisms among groups of order $n$?

The number of automorphisms of the group $$C_n$$ is $$|\text{Aut}(C_n)|= \varphi(n)$$, the number of generators of the group. Using GAP, I have checked that this is minimal among groups of order $$n$$ for every $$n<128$$, and it feels very plausible that this always holds.

However, I haven't managed to show this is the case; the fact that a group can have elements of high order fixed by every automorphism seems like an obstacle to several approaches. A proof, or references to a theorem implying this, would be welcome.

Googling “groups with small automorphism group” quickly led me to this write-up by Robert Wilson.

As stated in theorem 1, for finite abelian groups we have $$|\mathrm{Aut}(G)|\geq|\phi(|G|)|$$, with equality if and only if $$G$$ is cyclic.

As noted immediately after, whether the conclusion was true if we drop the assumption that $$G$$ is abelian was question 15.43 of the Kourovka Notebook, there attributed to M. Deaconescu.

He asked: (i) Does $$|\mathrm{Aut}(G)|\geq|\phi(|G|)|$$ hold for every finite group $$G$$? and (ii) If $$|\mathrm{Aut}(G)|=\phi(|G|)$$, must $$G$$ be finite cyclic?

The answer is “No” to both parts. Counterexamples to the first question (what you are asking), emerged from quasisimple groups in the Atlas. In particular, $$G=12M_{22}$$ (where $$M_{22}$$ is the Mathieu group on 22 letters) has $$\phi(|G|)/|G| = \frac{16}{77}$$, and $$|\mathrm{Aut}(G)|/|G|=\frac{1}{6}$$, which is smaller.

The full citation for the paper referenced in that write-up is:

J.N. Bray, R. A. Wilson. On the orders of automorphism groups of finite groups, Bull. London Math. Soc. 37 no. 3 (2005), pp. 381-385.