Submitted exclusively to the London Mathematical Society DOI: 10.1112/S0000000000000000

ON THE ORDERS OF AUTOMORPHISM GROUPS OF FINITE GROUPS

JOHN N. BRAY and ROBERT A. WILSON

Abstract

In the Kourovka Notebook, Deaconescu asks if |Aut G| ⩾ ϕ(|G|) for all finite groups G, where ϕ denotes the Euler totient function; and whether G is cyclic whenever |Aut G| = ϕ(|G|). We answer both questions in the negative. Moreover we show that |Aut G|/ϕ(|G|) can be made arbitrarily small.

1. The question, and some answers

Conventions. Throughout this paper, we shall only consider finite groups. The notation for group structures is based on that used in the Atlas [2]. The notation O_{2}(G), O_{2′}(G) and Aut G is standard.

Let ϕ denote the Euler totient function, so that ϕ(n) is the number of integers m with 1 ⩽ m ⩽ n such that m and n are coprime, and

\frac{ϕ(n)}{n} = ∏_{i=1}^{r} \frac{p_{i} − 1}{p_{i}},

where p_{1} < p_{2} < … < p_{r} are the prime factors of n. It is easy to see that for finite abelian groups G, we have |Aut G| ⩾ ϕ(|G|), with equality if and only if G is cyclic. In Problem 15.43 of the Kourovka Notebook [3], Deaconescu asks if the same is true for arbitrary finite groups G. More specifically:

Let G be a finite group of order n.

a) Is it true that |Aut G| ⩾ ϕ(n) where ϕ is Euler’s function?

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

In this note we show that the answer to both questions is no. Indeed, we shall prove:

Main Theorem. For all ε > 0 there exists a group G such that |Aut G| < ε.ϕ(|G|).

In the course of this paper, it will transpire that there are infinitely many groups G satisfying |Aut G| < ϕ(|G|), and infinitely many non-cyclic groups G which satisfy |Aut G| = ϕ(|G|).

2000 Mathematics Subject Classification 20E36 (primary), 20F28 (secondary).

2 john n. bray and robert a. wilson

Part (a)

Examining some quasisimple groups, we quickly found that the perfect groups G ≅ (3 × 4 × 2)·L_{3}(4), 12·M_{22} and (3^{2} × 4)·U_{4}(3) satisfy |Aut G| < ϕ(|G|), answering this part of the question in the negative. More precisely:

group G  primes dividing |G|  Aut G  \frac{|Aut G|}{|G|}  \frac{ϕ(|G|)}{|G|}
(3 × 4 × 2)·L_{3}(4)  2, 3, 5, 7  L_{3}(4):2^{2}  \frac{1}{6}  \frac{8}{35}
12·M_{22}  2, 3, 5, 7, 11  M_{22}:2  \frac{1}{6}  \frac{16}{77}
(3^{2} × 4)·U_{4}(3)  2, 3, 5, 7  U_{4}(3):D_{8}  \frac{2}{9}  \frac{8}{35}

Part (b)

One non-cyclic group G satisfying |Aut G| = ϕ(|G|) is G ≅ 2 × 3 × 5 × 11 × M_{11}. Each of the five direct factors of G is characteristic, and we obtain

Aut G ≅ Aut 2 × Aut 3 × Aut 5 × Aut 11 × Aut M_{11} ≅ 1 × 2 × 4 × 10 × M_{11},

a group of order ϕ(|G|). More generally, if 2.3.5.11 = 330 | m, then the non-cyclic group G ≅ C_{m} × M_{11} has automorphism group Aut G ≅ (Aut C_{m}) × M_{11}, and is thus a group of order ϕ(|G|).

The smallest non-cyclic group G we know of that satisfies |Aut G| = ϕ(|G|) is a group of order 56448, namely the group G ≅ 2^{4}:L_{3}(2)×3×7, where the 2^{4}:L_{3}(2) is isomorphic to the centraliser of an involution in M_{23}. In the direct factor 2^{4}:L_{3}(2), when we consider the normal 2^{4} as an 𝔽_{2}-module for a complementary L_{3}(2), it is uniserial with a single non-zero proper submodule; this submodule has dimension 1. We have Aut G ≅ 2^{3}:L_{3}(2) × 2 × 6. We may take G to be the group ⟨g_{1}, g_{2}, g_{3}⟩ where:

\begin{matrix}g_{1} = & (1, 3)(2, 4)(5, 15, 6, 16)(7, 9)(8, 10)(11, 13, 12, 14), \\ g_{2} = & (1, 3, 5)(2, 4, 6)(7, 15, 11)(8, 16, 12), \\ g_{3} = & (17, 18, 19)(20, 21, 22, 23, 24, 25, 26). \end{matrix}

In fact, this group G is a special case of the groups we construct in the next section.

2. Proof of the Main Theorem

Let P be a non-empty finite set of primes such that p ≡ 7 (mod 8) for all p ∈ P. We shall consider groups of the form

G = C × ∏_{p∈P} M_{p},

where C is a cyclic subgroup of odd order and M_{p} is a perfect group of shape 2^{f(p)}:L_{2}(p). (We shall define the groups M_{p} below.)

Lemma 1. For all p ∈ P, the subgroups C and M_{p} are characteristic in G. Thus

Aut G ≅ Aut C × ∏_{p∈P} Aut M_{p}.

on the orders of automorphism groups of finite groups 3

Proof. First, note that C = O_{2′}(G), so is characteristic in G. Let N = F(G), the Fitting subgroup of G, so that N is also characteristic in G. Now G/N ≅ ∏_{p∈P} L_{2}(p) is a direct product of non-isomorphic simple groups. For each p ∈ P, G/N has a unique normal subgroup N_{p}/N such that N_{p}/N ≅ L_{2}(p). Thus N_{p}/N is characteristic in G/N, and since N is characteristic in G, we get (for each p ∈ P) that N_{p} is a characteristic subgroup of G. In fact, N_{p} = ⟨N, M_{p}⟩, and we have

N_{p} = C × M_{p} × ∏_{q∈P \{p}} O_{2}(M_{q}).

Since C and O_{2}(M_{q}) are abelian and M_{p} is perfect, we obtain N′_{p} = M_{p}, and thus M_{p} is also a characteristic subgroup of G.

Since C is cyclic, Aut C is an abelian group of order ϕ(|C|). We now concentrate on the groups M_{p} (where we may now fix p ≡ 7 (mod 8)). The order of Aut M_{p} depends crucially on the structure of O_{2}(M_{p}) as an 𝔽_{2}L_{2}(p)-module. We aim to construct a uniserial module with composition factors of dimensions 1 and \frac{1}{2}(p − 1), so that M_{p} has centre of order 2, in such a way that M_{p} has trivial outer automorphism group. It will then follow that |Aut M_{p}| = \frac{1}{2}|M_{p}|. In what follows, we shall use V_{1} · V_{2} to denote a non-split extension of modules V_{1} by V_{2} where V_{1} is the submodule and V_{2} is the quotient.

Lemma 2. Let H denote the simple group L_{2}(p), where p ≡ 7 (mod 8), and let V denote the permutation module over 𝔽_{2} of H on the p + 1 cosets of the Borel subgroup. Then V is isomorphic to the trivial PIM (projective indecomposable module) for H, and has structure 1 · (U_{1} ⊕ U_{2}) · 1, where U_{1} and U_{2} are absolutely irreducible.

Proof. First we need some notation for some elements of H. We use t to denote any element of order p; all elements of order p in H are conjugate to t or t^{−1}. The other elements of H have order dividing \frac{1}{2}(p − 1) or \frac{1}{2}(p + 1); all such non-identity elements can be notated by x, y or z where:

x ≠ 1 has order dividing \frac{1}{2}(p − 1). y ≠ 1 has order dividing \frac{1}{4}(p + 1). z ≠ 1 has order dividing \frac{1}{2}(p + 1), but does not have order dividing \frac{1}{4}(p + 1).

Four ordinary irreducible characters of L_{2}(p) are

\begin{matrix}Element & 1 & x & y & z & t & t^{−1} \\ χ_{0} & 1 & 1 & 1 & 1 & 1 & 1 \\ χ_{1} & \frac{1}{2}(p − 1) & 0 & −1 & 1 & β & γ \\ χ_{2} & \frac{1}{2}(p − 1) & 0 & −1 & 1 & γ & β \\ χ_{3} & p & 1 & −1 & −1 & 0 & 0 \end{matrix}

where β and γ denote the irrationalities \frac{1}{2}(−1 ± \sqrt{−p}); thus β and γ have minimal polynomial X^{2}+X+\frac{1}{4}(p+1). Since χ_{0}, χ_{1} and χ_{2} remain irreducible on restriction to the Borel subgroup of shape p:(\frac{p−1}{2}) (which has odd order because p ≡ 3 (mod 4)), they remain irreducible on reduction modulo 2. Moreover, since p ≡ 7 (mod 8), X^{2} + X + \frac{1}{4}(p + 1) reduces modulo 2 to X^{2} + X, which has roots in 𝔽_{2}, so the 4 john n. bray and robert a. wilson

corresponding representations can be written over 𝔽_{2}. For i ∈ {0, 1, 2} let φ_{i} denote the 2-modular Brauer character which is the restriction of χ_{i} to 2-regular classes, and let U_{i} be a module affording the character φ_{i}.

Now let P(1) denote the PIM of the trivial representation in characteristic 2, and let Perm(p + 1) denote the characteristic 2 permutation module of L_{2}(p), of degree p + 1, on the cosets of the Borel subgroup p:(\frac{p−1}{2}). Since p:(\frac{p−1}{2}) has odd order, Perm(p + 1) is projective, and thus, since it contains a trivial submodule, contains a copy of P(1). Since the corresponding characteristic 0 permutation module of degree p + 1 has character χ_{0} + χ_{3} and any element z is necessarily 2-singular, we see that Perm(p + 1) has Brauer character 2φ_{0} + φ_{1} + φ_{2}, and thus composition factors U_{0}, U_{0}, U_{1}, U_{2}.

Now P(1) has a unique simple submodule, and unique simple quotient, and thus (since 4 | |L_{2}(p)|) has the form 1 · U · 1 where U is a non-zero module. Since U_{1} and U_{2} are conjugate under an outer automorphism of L_{2}(p) while U_{0} = 1 remains invariant, we obtain that P(1) ≅ Perm(p + 1) ≅ 1 · (U_{1} ⊕ U_{2}) · 1. This structure is valid over any field of characteristic 2.

Since modules with simple socle M embed in the PIM corresponding to M, there are unique 𝔽_{2}L_{2}(p)-modules of shapes 1 · U_{1} and 1 · U_{2} while there are no 𝔽_{2}L_{2}(p)- modules of shapes 1 · U_{1} · 1 or 1 · U_{2} · 1. We now define M_{p} ≅ 2^{f(p)}:L_{2}(p) to be the split extension of the 𝔽_{2}L_{2}(p)-module 1 · U_{1} by L_{2}(p); in particular f(p) = \frac{1}{2}(p + 1). It remains to prove:

Lemma 3. With this definition, M_{p} has trivial outer automorphism group.

Proof. Since O_{2}(M_{p}) is a characteristic subgroup of M_{p}, any automorphism of M_{p} permutes the complements to O_{2}(M_{p}) in M_{p}. Now let S denote a complementary L_{2}(p) in M_{p}. We have ensured that the module 1 · U_{1} has zero 1-cohomology; thus M_{p} has just one class of complements, so we may assume that our automorphism α of M_{p} normalises S. Since the module 1 · U_{1} is not invariant under outer automorphisms of S, α|_{S} must be an inner automorphism of S, and adjusting by an inner automorphism of M_{p} that is conjugation by an element of S, we may assume that α centralises S. So now α is an 𝔽_{2}S-module automorphism of 1 · U_{1}, and thus is a scalar, and therefore trivial.

It follows that |Aut M_{p}| = \frac{1}{2}|M_{p}| and therefore

\frac{|Aut G|}{|G|} = \frac{ϕ(|C|)}{|C|} × \frac{1}{2^{|P |}}

whence

\frac{|Aut G|}{ϕ(|G|)} = \frac{ϕ(|C|)}{|C|} × \frac{|G|}{ϕ(|G|)} × \frac{1}{2^{|P |}} .

Now if for all p ∈ P and odd prime divisors q of p(p + 1)(p − 1) we have that q | |C| then it follows that ϕ(|C|)/|C| = 2ϕ(|G|)/|G|, and therefore

\frac{|Aut G|}{ϕ(|G|)} = \frac{ϕ(|C|)}{|C|} × \frac{|G|}{ϕ(|G|)} × \frac{1}{2^{|P |}} = \frac{1}{2^{|P |−1}} .

In particular, this holds if |C| is the product of all the odd primes which divide p(p + 1)(p − 1) for some p ∈ P. To complete the proof of the Main Theorem, we invoke Dirichlet’s Theorem that there are infinitely many primes p such that p ≡ 7 (mod 8) to conclude that |P | can be made arbitrarily large.

3. Further work

We have now extended our constructions and have been able to show that the Main Theorem holds when G is restricted to being soluble, and also when G is restricted to being perfect. Moreover, there are infinitely many perfect groups G and infinitely many non-cyclic soluble groups G such that |Aut G| = ϕ(|G|). These results are the subject of a forthcoming publication [1].

References

1. J. N. Bray and R. A. Wilson. On the orders of automorphism groups of finite groups. II. In preparation.

2. J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson. An 𝔸𝕋𝕃𝔸𝕊 of Finite Groups. Clarendon Press, Oxford (1985; reprinted with corrections, 2003).

3. E. I. Khukhro and V. D. Mazurov (Eds). Unsolved problems in group theory. The Kourovka Notebook, no. 15. Novosibirsk, 2002.

John N. Bray and Robert A. Wilson. School of Mathematics and Statistics, University of Birmingham, Edgbaston, Birmingham, B15 2TT.

jnb@maths.bham.ac.uk R.A.Wilson@bham.ac.uk