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

ON THE ORDERS OF AUTOMORPHISM GROUPS OF FINITE GROUPS. II

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|). In an earlier paper we have answered both questions in the negative, and shown that |Aut G|/ϕ(|G|) can be made arbitrarily small. Here we show that these results remain true if G is restricted to being perfect, or soluble.

1. The question, and general overview

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 [1] we showed that neither statement holds for arbitrary finite groups, thus solving Problem 15.43 of the Kourovka Notebook [5].

On the other hand they hold (trivially) for finite simple groups (indeed for all finite groups with trivial centre), and one is led to ask: For what classes of finite groups do the statements hold?

A long-standing conjecture of Schenkman [6], that if G is a finite non-cyclic pgroup of order at least p^{3} then |G| divides |Aut G|, would imply that both statements hold for finite nilpotent groups. Indeed, this is known to hold for nilpotent groups of class 2, see Schenkman [6].

In this paper we show that the statements do not hold for the class of perfect groups, nor for the class of soluble groups. As in [1], we actually prove stronger results:

Theorem 1. For all ε > 0 there exists a finite perfect group G such that |Aut G| < ε.ϕ(|G|).

Theorem 2. For all ε > 0 there exists a finite soluble group G such that |Aut G| < ε.ϕ(|G|).

Theorem 3. For all N ∈ ℕ there exists a finite perfect group G with |G| > N such that |Aut G| = ϕ(|G|).

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

Theorem 4. For all N ∈ ℕ there exists a finite non-cyclic soluble group G with |G| > N such that |Aut G| = ϕ(|G|).

We were unable to resolve the case of supersoluble groups, but are marginally inclined to the view that:

Conjecture. If G is a finite non-nilpotent supersoluble group, then |Aut G| > ϕ(|G|).

Conventions. Throughout this paper, we shall only consider finite groups. The notation for group structures is based on that used in the Atlas [3]. The notation O_{p}(G), O_{p′}(G), O^{p}(G), Aut G, and Out(G) is standard. The abbreviation PIM stands for projective indecomposable module. If U and V are modules then U · V denotes a non-split extension of U by V with U being the submodule and V being the quotient.

2. Some modules and cohomology

We need some information about modules and cohomology of L_{2}(p), especially when p ≡ 7 (mod 8). The following information was established in [1]:

Lemma 5. For p prime and p ≡ 7 (mod 8) there are precisely two isomorphism classes of 𝔽_{2}L_{2}(p)-modules 1 · U in which U is absolutely irreducible of dimension \frac{1}{2}(p − 1), and the 1 denotes the trivial module. These two modules are interchanged by the non-trivial outer automorphism of L_{2}(p), and both of these modules have zero 1-cohomology. These two modules have the forms 1 · U_{1} and 1 · U_{2} where U_{1} and U_{2} are not isomorphic.

For all primes p there are just p irreducible modules of SL_{2}(p) in characteristic p. Their dimensions are all different, and at most p, and we label the SL_{2}(p)-irreducible of dimension i (1 ⩽ i ⩽ p) as V_{i}. For p odd, the central involution of SL_{2}(p) acts trivially on V_{i} if and only if i is odd; in such cases we regard V_{i} as being an L_{2}(p) module. Of course, V_{1} is the trivial module for L_{2}(p).

For p ≡ 3 (mod 4) the Brauer tree of the principal block of L_{2}(p) in characteristic p is a straight line with \frac{1}{2}(p + 1) nodes and diagram

[figure: 1; p − 1; p + 1; p − 1; p + 1; \frac{1}{2}(p − 1); V_{1}; V_{p−2}; V_{3}; V_{\frac{1}{2} (p−1)}]

where we have labelled the nodes with the degrees of ordinary characters to which they correspond and we have labelled the edges with their corresponding p-modular irreducibles. From the Brauer tree one reads off the PIMs

V_{1} · V_{p−2} · V_{1} and V_{p−2} · (V_{1} ⊕ V_{3}) · V_{p−2}

for all primes p ⩾ 7 with p ≡ 3 (mod 4). (In fact, these PIM structures are valid for all primes p ⩾ 5.) Note that the V_{i} and all of the PIMs for L_{2}(p) (and also SL_{2}(p)) can be realised over 𝔽_{p}.

Let W be the 𝔽_{p}L_{2}(p)-module (V_{1} ⊕ V_{3}) · V_{p−2} (with simple head). So W is a

on the orders of automorphism groups of finite groups. ii 3

quotient of the PIM V_{p−2} · (V_{1} ⊕ V_{3}) · V_{p−2} and therefore is unique. One can also read off from the PIMs that W has zero 1-cohomology whenever p ⩾ 7.

For p prime and p ≡ 7 (mod 8), we define J_{p} to be J_{p} ≅ (2^{(p+1)/2}×p^{p+2}):L_{2}(p), in which the complementary L_{2}(p) act on O_{p}(J_{p}) as the module W ≅ (1⊕V_{3})·V_{p−2} and on O_{2}(J_{p}) as the module 1 · U_{1} of Lemma 5. The groups J_{p}/O_{p}(J_{p}) are isomorphic to the groups M_{p} we constructed in [1].

3. Perfect groups

In this section, we construct infinite series of finite perfect groups which prove Theorems 1 and 3. We let r ⩾ 11 be a prime, and define G to be the direct product of certain perfect groups B_{p} for each prime p between 3 and r inclusive:

G = ∏_{p∈π} B_{p} = ∏_{p=3, p prime}^{r} B_{p},

where π is the set of odd primes not exceeding r. Firstly, we take B_{3} ≅ 3^{6}:M_{11}, where O_{3}(B_{3}) when regarded as an 𝔽_{3}M_{11}-module is a uniserial module of shape 1 · 5a (this module is isomorphic to the unique 6-dimensional submodule of the 𝔽_{3}-permutation module of M_{11} on the 12 cosets of L_{2}(11)). Note also that the composition factor 5a is absolutely irreducible. We have:

Lemma 6. The 𝔽_{3}M_{11}-module 1 · 5a has zero 1-cohomology.

Proof. This is an easy calculation using Magma [2]. Alternatively, the trivial module has zero 1-cohomology since M_{11} is perfect, and it can be shown that the module 5a does not occur in the second Loewy layer of the trivial PIM. Thus both composition factors of 1 · 5a have trivial 1-cohomology, and so does the whole module.

Lemma 7. If p = 3, so that B_{p} = B_{3} ≅ 3^{6}:M_{11}, then B_{p} = B_{3} has outer automorphism group of order 2. Thus |Aut B_{p}| = \frac{2}{3}|B_{p}| = \frac{p−1}{p}|B_{p}|.

Proof. Since O_{3}(B_{3}) is a characteristic subgroup of B_{3}, any automorphism of B_{3} permutes the complements to O_{3}(B_{3}) in B_{3}. Now let S denote a complementary M_{11} in B_{3}. Since we have ensured that the 𝔽_{3}M_{11}-module O_{3}(B_{3}) has zero 1cohomology, we may assume our automorphism, α say, normalises S. But M_{11} has trivial outer automorphism group, and adjusting α by an inner automorphism that is conjugation by an element of S, we may assume that α centralises S. So now α is an 𝔽_{3}S-module automorphism of O_{3}(B_{3}) ≅ 1 · 5a, and is thus a non-zero scalar. There are two of these and so |Out B_{p}| = 2. In fact, Aut B_{p} ≅ 3^{5}:(M_{11} × 2).

For p ⩾ 5 and p ≢ 7 (mod 8), we take B_{p} ≅ p_{+}^{1+2}:SL_{2}(p). For p ⩾ 5 and p ≡ 7 (mod 8), we take B_{p} ≅ p_{+}^{1+2}:SL_{2}(p) or B_{p} ≅ J_{p} ≅ (2^{(p+1)/2} × p^{p+2}):L_{2}(p), the group we constructed in Section 2 (we are free to choose either; this choice is necessary in order to prove Theorem 3).

The group p_{+}^{1+2}:SL_{2}(p) has a centre of order p, and is isomorphic to a vector stabiliser in the natural representation of Sp_{4}(p).

Lemma 8. Let H = N:K and let K act faithfully on N. Suppose α ∈ Aut H centralises N and normalises K. Then α centralises K. (So α = 1.)

Proof. For all g ∈ N, k ∈ K we have (g^{k})α = g^{k} since g^{k} ∈ N. On the other hand (g^{k})α = (gα)^{(kα)} = g^{(kα)}. So for all g ∈ N, k ∈ K we have g^{k(kα)^{−1}} = g, whence kα = k since K acts faithfully on N.

Lemma 9. For all primes p ⩾ 5 we have Aut(p_{+}^{1+2}:SL_{2}(p)) ≅ p^{2}:GL_{2}(p). So if B_{p} ≅ p_{+}^{1+2}:SL_{2}(p) we have |Aut B_{p}| = \frac{p−1}{p}|B_{p}|.

Proof. The 1-space stabiliser in Sp_{4}(p) is a group p_{+}^{1+2}:GL_{2}(p) which induces a group p^{2}:GL_{2}(p) of automorphisms on its normal subgroup p_{+}^{1+2}:SL_{2}(p).

The group p_{+}^{1+2}:SL_{2}(p) contains exactly p^{2} involutions, which are permuted faithfully by the above group p^{2}:GL_{2}(p). Each of these involutions has centraliser of shape p × SL_{2}(p), and these define the p^{2} complements SL_{2}(p) [by taking the O^{p} or the derived subgroup]. Moreover, these involutions generate p_{+}^{1+2}:2, and support a natural affine plane structure; three involutions are collinear in this affine plane if and only if they generate a subgroup isomorphic to D_{2p}.

We already see the full automorphism group p^{2}:GL_{2}(p) of this affine plane, so the only way the automorphism group of p_{+}^{1+2}:SL_{2}(p) could be any bigger is if there were a non-trivial kernel, i.e. an automorphism centralising all p^{2} involutions. Such an automorphism would have to normalise, and therefore by Lemma 8 centralise, each of the complements, as well as centralising the group p_{+}^{1+2}:2 generated by the involutions. Therefore it is the trivial automorphism on each complementary SL_{2}(p), and hence on p_{+}^{1+2}:SL_{2}(p), and the lemma follows.

Lemma 10. If p ≡ 7 (mod 8) and H is the group J_{p} ≅ (2^{(p+1)/2} × p^{p+2}):L_{2}(p) we constructed in Section 2, then Aut H ≅ (2^{(p−1)/2} × p^{p+1}):(L_{2}(p) × C_{p−1}). So if B_{p} ≅ J_{p} we have |Aut B_{p}| = \frac{1}{2} \frac{p−1}{p}|B_{p}|.

Proof. The elementary abelian subgroups O_{2}(H) and O_{p}(H) are characteristic in H; therefore K := O_{2}(H) × O_{p}(H) is also characteristic in H. Since O_{2}(H) and O_{p}(H) both have zero 1-cohomology as L_{2}(p)-modules (see Section 2), H has just one conjugacy class of complementary subgroups L_{2}(p). So let α ∈ Aut H and let S be a complementary subgroup L_{2}(p). Modulo inner automorphisms, α normalises S. Now O_{2}(H) when regarded as an 𝔽_{2}S-module does not admit the non-trivial outer automorphism of S ≅ L_{2}(p), see Lemma 5. So α induces an inner automorphism when restricted to S, and adjusting by an inner automorphism of H that is conjugation by an element of S, we may assume that α centralises S. So now α induces an 𝔽_{2}S-module automorphism on O_{2}(H) and an 𝔽_{p}S-module automorphism on O_{p}(H), and both of these are scalars. Since H ≅ J_{p} has centre of order 2p, the result follows.

Lemma 11. For all primes p with 3 ⩽ p ⩽ r, the groups B_{p} are characteristic in G.

Proof. Let π be the set of all primes between 3 and r inclusive. Let N = F(G), the Fitting subgroup of G, so that N is characteristic in G. Then

G/N ≅ ∏_{p∈π} S_{p},

where S_{3} ≅ M_{11} and S_{p} ≅ L_{2}(p) whenever p ⩾ 5. So G/N has a unique normal subgroup N_{p}/N such that N_{p}/N ≅ S_{p}, and for all p we get that N_{p} is characteristic in G. In fact

N_{p} = B_{p} × ∏_{q∈π′} O_{{2,q}}(B_{q}),

where π′ = π \ {p}, with the O_{{2,q}}(B_{q}) being nilpotent groups of class at most 2. Therefore B_{p} = N′′_{p} is a characteristic subgroup of G.

Since all of the B_{p} are characteristic in G, we have Aut G ≅ ∏_{p∈π} Aut B_{p}. We have also established for all p ∈ π that |Aut B_{p}| = \frac{p−1}{p}|B_{p}| or \frac{1}{2} \frac{p−1}{p}|B_{p}|, with the latter case occurring if and only if B_{p} ≅ J_{p}. Therefore we have

\frac{|Aut G|}{|G|} = \frac{1}{2^{m}} × ∏_{p∈π} \frac{p − 1}{p},

where m is the number of primes p > 3 for which B_{p} ≅ J_{p}. Now when p > 3 the largest prime divisor of |B_{p}| is p, and the primes dividing |B_{3}| are 2, 3, 5 and 11. But r ⩾ 11, so the primes dividing |G| are precisely the primes that do not exceed r, and thus we have

|Aut G| = \frac{1}{2^{m−1}} × \frac{1}{2} × ∏_{p∈π} \frac{p − 1}{p} × |G| = \frac{1}{2^{m−1}} × ϕ(|G|).

When m = 1 this gives |Aut G| = ϕ(|G|). We now invoke Dirichlet’s Theorem that there are infinitely many primes p with p ≡ 7 (mod 8) to complete the proofs of Theorems 1 and 3.

Remark. It is convenient but not essential to take all odd primes up to r in the definition of G. But every odd prime dividing |G| must be one of these defining primes. To this end, let π be a set of odd primes such that 3, 5, 11 ∈ π and if p ∈ π then q ∈ π whenever q is an odd prime factor of p − 1 or p + 1. Let the B_{p} be as above. Then the group

G = ∏_{p∈π} B_{p}

satisfies |Aut G| = 2^{1−m}ϕ(|G|) where m is the number of p ∈ π such that B_{p} ≅ (2^{(p+1)/2} × p^{p+2}):L_{2}(p).

4. Soluble groups

In this section, we construct infinite series of finite soluble groups in order to prove Theorems 2 and 4.

We define B_{3} to be the unique group of shape 3_{+}^{1+2}:4 with centre of order 3; this group is SmallGroup(108,15) in various versions of Magma [2], including Version 2.10. Let π be a finite non-empty set of primes such that p ≡ 1 or 7 (mod 8) for all p ∈ π, and let π′ = π ∪ {3}. We shall consider the groups:

G = ∏_{p∈π′} B_{p} ≅ (∏_{p∈π} p_{+}^{1+2}:2·S_{4}^{−}) × 3_{+}^{1+2}:4,

where B_{p} ≅ p_{+}^{1+2}:2·S_{4}^{−} for all p ∈ π and 2·S_{4}^{−} denotes the proper double cover of S_{4} in which transpositions do not lift to involutions.

It is well-known, see Dickson [4], that L_{2}(p) contains subgroups isomorphic to S_{4} if and only if p ≡ ±1 (mod 8). Such a subgroup is unique up to conjugacy in PGL_{2}(p), and is also self-normalising in PGL_{2}(p). Now the double cover SL_{2}(p) of L_{2}(p) contains just one involution, namely its central one, and so the preimage 2.S_{4} in SL_{2}(p) is 2·S_{4}^{−}. Thus the group B_{p} ≅ p_{+}^{1+2}:2·S_{4}^{−} exists and has centre of order p. Note that the primes dividing |G| are precisely those in the set {2, 3} ∪ π.

Lemma 12. For all p ∈ π, the groups B_{p} are characteristic in G.

Proof. Let π′′ := (π ∪ {3}) \ {p} = π′ \ {p}. The group O_{2′}(G) is the direct product of the subgroups W_{p} := O_{p}(B_{p}) ≅ p_{+}^{1+2}. We have

O_{{2,p}′}(G) = O_{p′}(O_{2′}(G)) = ∏_{q∈π′′} W_{q}.

Thus we calculate that

H := C_{G}(O_{p′}(O_{2′}(G))) = B_{p} × ∏_{q∈π′′} Z(W_{q}) ≅ p_{+}^{1+2}:2·S_{4}^{−} × ∏_{q∈π′′} C_{q}.

Now B_{p} is characteristic in H, since it is the subgroup generated by the elements of order 4. Since H is a characteristic subgroup of G we conclude that B_{p} is also.

Lemma 13. The group B_{3} is also characteristic in G.

Proof. This proof is very similar to the proof of Lemma 12, and we use the notation W_{q} from the proof of that lemma here. The group

H := C_{G}(O_{3′}(O_{2′}(G))) = B_{3} × ∏_{q∈π} Z(W_{q}) ≅ 3_{+}^{1+2}:4 × ∏_{q∈π} C_{q}

is a characteristic subgroup of G. Now B_{p} is characteristic in H, since it is the subgroup generated by the elements of order 4. Since H is a characteristic subgroup of G we conclude that B_{p} is also.

Lemma 14. For all p ∈ π, we have Aut B_{p} ≅ p^{2}:(2·S_{4}^{−} ∘ C_{p−1}). So for all such primes p we have |Aut B_{p}| = \frac{1}{2} \frac{p−1}{p}|B_{p}|.

Proof. We adapt the proof of Lemma 9. First note that H = B_{p} ≅ p_{+}^{1+2}:2·S_{4}^{−} embeds in p_{+}^{1+2}:GL_{2}(p), in which its normaliser is p_{+}^{1+2}:(2·S_{4}^{−}∘C_{p−1}), and therefore its automorphism group contains p^{2}:(2·S_{4}^{−} ∘ C_{p−1}).

Note that the latter group acts transitively and faithfully on the p^{2} involutions in H, so if we have any further automorphism α, we may assume α fixes one of these involutions, say z. Therefore α fixes C_{H}(z) ≅ p × 2·S_{4}^{−}, and therefore normalises the corresponding subgroup K = O^{p}(C_{H}(z)) ≅ 2·S_{4}^{−} of H.

Now both α and K act on the affine plane defined by the p^{2} involutions (as in

on the orders of automorphism groups of finite groups. ii 7

Lemma 9), and both fix the same point, which we can regard as the origin. In the resulting action on the vector space of order p^{2}, the image of α normalises the image of K inside GL_{2}(p) (indeed, K:⟨α⟩ acts on this vector space). But this action of K is faithful and N_{GL_{2}(p)}(2·S_{4}^{−}) ≅ 2·S_{4}^{−} ∘ C_{p−1}. But we have already seen a group p^{2}:(2·S_{4}^{−} ∘C_{p−1}) of automorphisms of H acting faithfully on the affine plane, and so we may assume that α acts trivially on the affine plane. In other words, α centralises all p^{2} involutions in H, so centralises the group p_{+}^{1+2}:2 which they generate.

We now know that α centralises p_{+}^{1+2}, and normalises a complementary 2·S_{4}^{−}. Therefore, by Lemma 8, α is the trivial automorphism of H. This completes the proof of the lemma.

An easy calculation gives Aut B_{3} ≅ 3^{2}:SD_{16}. Since all of the B_{p} are characteristic in G, we obtain Aut G ≅ ∏_{p∈π∪{3}} Aut B_{p}. Therefore we have

\frac{|Aut G|}{|G|} = \frac{4}{3} × (\frac{1}{2})^{|π|} × ∏_{p∈π} \frac{p − 1}{p} = \frac{1}{2^{|π|−2}} × \frac{ϕ(|G|)}{|G|},

and so

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

When |π| = 2 this gives |Aut G| = ϕ(|G|). Dirichlet’s Theorem tells us that π can be made arbitrarily large, thus proving Theorem 2, and also that there are infinitely many size 2 possibilities for π, thus proving Theorem 4.

Remark. In the above construction for G we can replace the group B_{3} by the cyclic group of order 3, in which case |Aut G| = 2^{1−|π|}ϕ(|G|). However the proofs are slightly different. The smallest non-cyclic group G we know of that satisfies |Aut G| = ϕ(|G|) is now the group G ≅ 3 × 7_{+}^{1+2}:2·S_{4}^{−} of order 49392, narrowly beating the example 2^{4}:L_{3}(2) × 3 × 7 of order 56448 that we gave in [1].

Acknowledgements. We are grateful to Chris Parker for helping us simplify some of the proofs in this paper.

References

1. J. N. Bray and R. A. Wilson. On the orders of automorphism groups of finite groups. Bull. London Math. Soc., to appear.

2. J. J. Cannon et al. The Magma programming language, Version 2.10. School of Mathematics and Statistics, University of Sydney (2003).

3. 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).

4. L. E. Dickson. Linear Groups, with an Exposition of the Galois Field Theory. Teubner, Leipzig (1901), reprinted Dover, New York (1958).

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

6. E. Schenkman. The existence of outer automorphisms of some nilpotent groups of class 2. Proc. Amer. Math. Soc. 6 (1955), 6–11.

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