Papers on Automorphism Groups

These papers construct finite groups with small automorphism groups. In Problem 15.43 of the Kourovka Notebook, Deaconescu asks if there is a finite group G such that |Aut G| < φ(|G|), and whether G is necessarily cyclic when |Aut G| = φ(|G|). The first paper shows that the answer to both questions is no, constructs infinitely many non-cyclic groups G with |Aut G| = φ(|G|), and shows that |Aut G| / φ(|G|) can be made arbitrarily small. The second paper shows that the results of the first paper remain valid when G is restricted to being perfect, or soluble.

J.N.Bray and R.A.Wilson. On the orders of automorphism groups of finite groups.
To appear Bull. London Math. Soc.
Preprint 2003/22 at the University of Birmingham.
Available as: dvi-format [22 K], pdf-format [166 K]. (5 pages.)

J.N.Bray and R.A.Wilson. On the orders of automorphism groups of finite groups. II.
Submitted to Bulletin/Journal/Proceedings of the LMS.
Preprint 2004/14 at the University of Birmingham.
Available as: dvi-format [37 K], pdf-format [203 K]. (7 or 8 pages.)

Smallest known non-cyclic groups with |Aut G| = φ(|G|)

This table provides the smallest known groups G for which |Aut G| = φ(|G|) as they come to our attention. A star (*) by the group name means that the group was [at one time] the smallest known group of its type, but never the smallest known such group.
We do not know of any supersoluble [or nilpotent] examples of such groups.
The Magma files need to be run in Versions 2.8 or later because the AutomorphismGroup functionality does not exist in earlier versions.

Group G |G| Aut G Magma file(s) Notes
3 × 71+2:2S4 49392 = 24.32.73 2 × 72:(2S4 × 3) Perm346; PC-Group Smallest known soluble example
24:L3(2) × 3 × 7 56448 = 27.32.72 23:L3(2) × 2 × 6 Perm26 Smallest known insoluble example
M11 × 2 × 3 × 5 × 11 2613600 = 25.33.52.112 M11 × 1 × 2 × 4 × 10 Perm32
51+4:6.A6 (*) 6750000 = 24.33.56 54:2.A6.4 Perm3143 Smallest known perfect example

Smallest known groups with |Aut G| < φ(|G|)

This table provides the smallest known groups G for which |Aut G| < φ(|G|) as they come to our attention. A star (*) by the group name means that the group was [at one time] the smallest known group of its type, but never the smallest known such group. We have also included all quasi-simple groups with this property.
We do not know of any supersoluble [or nilpotent] examples of such groups.
The Magma files need to be run in Versions 2.8 or later because the AutomorphismGroup functionality does not exist in earlier versions.

The first three groups were obtained by trawling the Holt-Plesken Magma perfect groups database. They are still being investigated.

Group G |G| Aut G Magma file(s) Notes
[28].L3(2) 43008 = 211.3.7 23a+3b:L3(2) [??]
[28].L3(2) 43008 = 211.3.7 23a.3b:L3(2) [??]
[28].L3(2) 43008 = 211.3.7 23a.3b:L3(2) [??]
(3 × 4 × 2).L3(4) 483840 = 29.33.5.7 L3(4):22
12.M22 5322240 = 29.33.5.7.11 M22:2
(4 × 32).U4(3) 117573120 = 29.38.5.7 U4(3).D8

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Last updated 24th June, 2004
Dr John N. Bray