On finite d$d$ ‐maximal groups

Let d$d$ be a positive integer. A finite group is called d$d$ ‐maximal if it can be generated by precisely d$d$ elements, whereas its proper subgroups have smaller generating sets. For d∈{1,2}$d\in \lbrace 1,2\rbrace$ , the d$d$ ‐maximal groups have been classified up to isomorphism and only partial results have been proved for larger d$d$ . In this work, we prove that a d$d$ ‐maximal group is supersolvable and we give a characterisation of d$d$ ‐maximality in terms of so‐called maximal (p,q)$(p,q)$‐pairs. Moreover, we classify the maximal (p,q)$(p,q)$ ‐pairs of small rank obtaining, as a consequence, the classification of the isomorphism classes of 3‐maximal finite groups.

a semidirect product of a cyclic group  of order  with a cyclic -group  and ∕  () has order .The structure of -maximal -groups has been investigated by Laffey [7].Adapting an argument of J. G. Thompson, he proved that, if  is an odd prime and  is a -maximal -group, then  has class at most 2 and the Frattini subgroup of  has exponent  and coincides with its derived subgroup; in particular, || ⩽  2−1 .The situation for  = 2 turned out to be much more intricate.In 1996, Minh [10] constructed a 4-maximal 2-group of class 3 and order 2 8 .Nowadays groups of order 2 8 can be examined using a computer.There are 20 241 groups  of order 2 8 with d() = 4, and only two of them are 4-maximal with nilpotency class 3.All the known -maximal 2-groups are of class at most 3 and the following question is open.
Clearly a nilpotent -maximal group must be a -group.In this paper, we are interested in maximal groups in the more general situation where  is not nilpotent.Using the classification of the finite non-abelian simple groups, we prove that a finite -maximal group is solvable and its order is divisible by at most two different primes, as the next result shows.
Theorem 1.3.Let  be a non-nilpotent -maximal group.Then there exist distinct primes  and  such that the derived subgroup  of  is a Sylow -subgroup of  and ∕ is a cyclic -group.Moreover, if  is a Sylow -subgroup of , then ∕  () has order .
In light of the previous result, it is natural to investigate the structure of finite -groups that can occur as the derived subgroup of a non-nilpotent -maximal group.We recall that a power automorphism of a finite group is an automorphism sending every subgroup to itself.If  is an elementary abelian -group, then a power automorphism of  is just scalar multiplication by some element of  ×  .
Definition 1.4.Let  and  be prime numbers.A maximal (, )-pair of rank  is a pair (, ) where  is a finite -group,  ∈ Aut() has prime order  dividing  − 1, and the following properties are satisfied: (a) the minimal number of generators of every subgroup of  is at most d() = ; (b) the image of  in Aut(∕Φ()) is a non-trivial power automorphism; (c) if  is a proper subgroup of  with d() = d(), then either () ≠  or the image of  in Aut(∕Φ()) is not a non-trivial power automorphism.
We reformulate Theorem 1.3 in terms of maximal pairs.Theorem 1.5.A finite group  is -maximal if and only if one of the following occurs: (1) the group  is a -maximal -group; (2) there exist a maximal (, )-pair (, ) of rank  − 1 and a cyclic -group ⟨⟩ such that  is isomorphic to  ⋊ ⟨⟩ and, for every  ∈ , one has () = ().
The Miller and Moreno classification of minimal non-cyclic groups can be essentially reformulated to saying that, if (, ) is a maximal (, )-pair of rank 1, then  has order .In Section 5, we classify the maximal (, )-pairs (, ) of rank 2, proving in particular that either  has exponent  and order at most  3 or (, ) = (3, 2), in which case there is a unique exceptional example with  of order 81 and class 3.This result, combined with a recent classification of the 3-maximal -groups [1], allows us to give in Section 5.1 the full classification of the finite 3-maximal groups.In Section 6, we classify the maximal (, )-pairs (, ) of rank 3: in this case,  has class at most 3 and order at most  6 , and if || =  6 , then (, ) = (3, 2).
The behaviour of maximal pairs of small rank suggests the following question.The following is our main contribution to the solution of the previous questions.It implies, in particular, that the derived length of a -maximal group of odd order is at most 3. Theorem 1.9.Let (, ) be a maximal (, )-pair.If  > 2, then  has class at most 2.
The proof of Theorem 1.9 is given in Section 4.2, and involves results on maximal pairs (, ) where  is regular (the definition of regularity is given in Section 4.1).The class of regular -groups is not only easier to study, but also a reasonable family to restrict to.Indeed, as soon as  ⩾ 2 and (, ) is a maximal pair of rank , the group  is regular (see Lemma 4.3).
Notation.We use standard group theory notation and write • Z() for the centre of , • Φ() for the Frattini subgroup of , • (  ()) ⩾1 for the lower central series of .
If  is a prime number,  a non-negative integer and  a finite -group, we write Ω  () and ℧  () for the following subgroups:

MAXIMAL GROUPS AND MAXIMAL PAIRS
In this section, we translate the problem of classifying -maximal groups into that of classifying maximal pairs, as defined in the Introduction.

2.1
The structure of -maximal groups The following theorem is [8, Cor.4].Its proof uses several different properties of the finite simple groups and requires their classification.The next result describes the -maximal groups with trivial Frattini subgroup.Note that, in the second case of Proposition 2.2, the subgroup  will necessarily act on the elementary abelian group  by scalar multiplication by elements of  ×  , and therefore, its order  will have to divide  − 1, yielding, in particular, that  is odd.Proposition 2.2.Let  be a -maximal finite group such that Φ() = 1.Then there exists a prime number  such that one of the following holds.
(1) The group  is an elementary abelian -group of rank .
(2) The group  is isomorphic to a semidirect product  ⋊ , where  is an elementary abelian -group of rank  − 1 and  is a central prime-order subgroup of GL −1 (  ).
Proof.If  is nilpotent, then  is a direct product of elementary abelian groups and (1) follows easily from -maximality.Let  = max ∈Syl() d() and observe that  < .From Theorem 2.1, we obtain a normal subgroup  of  such that ∕ is isomorphic to a semidirect product  ⋊ ⟨⟩ where  is elementary abelian of rank  − 1 and  acts on  as a non-trivial power automorphism.
In particular,  ≠ 2 and d(∕) = .We claim that  = 1.If this were not the case, since Φ() = 1, there would exist a maximal subgroup  of  such that  = , and thus, which is impossible.Let  ∈ ⟨⟩.If  acts non-trivially on , then d( ⋊ ⟨⟩) = , which gives ⟨⟩ = ⟨⟩.Since some Sylow subgroup of ⟨⟩ must act non-trivially on , we conclude that  has prime-power order, say   .Moreover,   must act trivially on .Since   ∈ Φ(⟨⟩), we conclude that   ∈ Φ() = 1.The proof is complete.□ The following two results are well known.The first is a direct consequence of the Schur-Zassenhaus theorem (see [11, 9.3.5]),whereas the second is [6, Thm.Lemma 2.4.Let  be a finite group,  an automorphism of  and  a normal -invariant subgroup whose order is coprime to the order of .Then C ∕ () = C  ()∕.Proposition 2.5.Let  be a -maximal finite group and assume that  is not a -group.Then  is isomorphic to a semidirect product  ⋊ ⟨⟩, where  is a -group for some odd prime , and  ∈ Aut() has prime-power order   for some  dividing  − 1.Moreover, d() =  − 1, and   centralises .

Maximal (𝒑, 𝒒)-pairs
Let  be a non-nilpotent -maximal group and let  and  be as in Proposition 2.5.In particular,   generates a central subgroup of  contained in Φ().It follows that the quotient ∕⟨  ⟩ is again -maximal and of order   , for some positive integer .Theorem 1.5 states that the study of these quotients is essentially equivalent to the investigation of maximal pairs.

Actions through characters
In this section, let  be a finite group and let  be an odd prime.Let  ∶  → ℤ ×  be a character.We define actions through characters and present some related results that we will apply in the study of maximal (, )-pairs.Definition 2.7.The group  is said to act on a group  through  if, for each  ∈  and g ∈ , one has g  = g () .Remark 2.8.Let (, ) be a maximal (, )-pair and  = ⟨⟩.Then, as a consequence of property (b) of maximal pairs, there exists a non-trivial character  ∶  → ℤ ×  such that  acts on ∕Φ() through .Moreover, it follows from property (c) that if  is a proper -invariant subgroup of  with d() = d() and such that  acts on ∕Φ() through a character   , then necessarily   = 1.
( Lemma 2.10.Let  be a finite -group that is also an -group and assume that the induced action of  on ∕ 2 () is through .Then, for all integers  ⩾ 1, the induced action of  on   ()∕ +1 () is through   .Lemma 2.11.Let  1 and  2 be finite -groups that are also -groups, and assume that  acts on  1 through .Moreover, let  ∶  1 →  2 be a surjective homomorphism respecting the action of , that is, for all  ∈  and g ∈  1 , one has that (g  ) = (g)  .Then  acts on  2 through .Lemma 2.12.Let  be a finite abelian -group on which  acts through .Assume that  = ⟨⟩ has order 2 and write Then  =  + ⊕  − .Lemma 2.13.Let  a finite -group on which  acts through .Let  be a normal -invariant subgroup of  such that the restriction of  to  equals the inversion map  ↦  −1 .Assume, moreover, that also the automorphism of ∕ that is induced by  is equal to the inversion map.Then,  is the inversion map on  and  is abelian.

GENERAL RESULTS ON MAXIMAL PAIRS
Until the end of Section 3, let (, ) denote a maximal (, )-pair of rank  and  = ⟨⟩.Moreover, let  ∶  → ℤ ×  be the character through which  acts on ∕Φ() as in Remark 2.8.
Proof.We start by proving (1).Applying Lemma 2.9(1) to  =  2 (), we assume without loss of generality that  is abelian of exponent dividing  2 .Then th powering is a homomorphism  → ℧ 1 (), and therefore, it follows from Lemma 2.11 that  acts on ℧ 1 () through .In order not to contradict property (c) of maximal pairs, the group  has to be equal to Ω 1 (), that is,  has exponent .Now (2) immediately follows from (1), whereas (3) is the combination of (1) with Lemma 2.10.□ The following result follows directly from Lemma 3.

THE STRUCTURE OF REGULAR PAIRS
In the wide world of -groups, the subclass of regular groups is somewhat tamer, sharing, in some sense, a number of properties with abelian groups.In this section, we study the effect of assuming regularity on a -group  that belongs to a maximal (, )-pair (, ).Moreover, we use regularity to prove general results on maximal pairs.

Regularity
Let  be a prime number and let  be a finite -group.Then,  is said to be regular if, for every ,  ∈ , one has ()  ≡     mod ℧ 1 ( 2 (⟨, ⟩)).
The following lemma collects the properties of regular groups we will make use of.We refer the interested reader to [5, Sec.III.10] for more on regularity.
Lemma 4.1.Let  be a prime number and  a finite -group.Let, moreover,  and  be non-negative integers and  and  be normal subgroups of .Then, the following hold.
(1) If the class of  is at most  − 1, then  is regular.
(2) If the exponent of  is , then  is regular.

Regular pairs
We now focus exclusively on regular pairs.Because of this, until the end of this section, let (, ) be a maximal regular (, )-pair of rank .The results proven here are not only interesting for their own sake, but will be also applied in the study of maximal pairs of small rank.
Proof.We work by induction on  and note that the case  = 3 is given by Lemma 4.6.Assume now that  > 3 and that the result holds for  − 1, in other words that  −1 () = ℧ 1 ( −3 ())  ().
Proof.Let  denote the class of : we work by induction on .The base of the induction is given by Lemma 4.6, so we assume that the result holds for  − 1, that is, that  3 () = ℧ 1 ()  ().We assume also, without loss of generality, that |  ()| =  and, for a contradiction, that   () is not contained in ℧ 1 (), that is, that ℧ 1 () ∩   () = 1.It follows from Theorem 1.9 and Lemma 4.

Contradiction. □
Corollary 4.9.Assume that the class of  is at least 3 and let  and  be positive integers.Then, the following hold.

MAXIMAL PAIRS OF RANK 2
In this section, we classify the maximal (, )-pairs of rank 2 and, as a consequence, the finite 3-maximal groups.To this end, until the end of Section 5, let (, ) be a maximal (, )-pair of rank 2. (1) an elementary abelian group of order  2 ; (2) an extraspecial group of order  3 and exponent .
It is easily seen that  = C  ( 2 ()) is abelian of order  3 .The rank of  being 2, this implies that ℧ 1 () =  3 (), and so,  is different from  = Ω 1 (), which is also a maximal subgroup of  (see Lemma 4.1( 7)).Since both  and  contain Φ(), both subgroups are -invariant.Write now  = ∕ 3 () and note that  is abelian and -invariant.Then, Lemma 2.12 implies that  =  + ⊕  − where both summands have order .Let  be the unique subgroup of  mapping to  − in .Since  acts on  3 () through  3 = , we derive from Lemma 2.13 that  is an elementary abelian subgroup of order  2 on which  acts through .This gives a contradiction to property (c) of maximal pairs and  = 2.
Proof.The claim is easily verified when || ⩽ 27, we assume therefore that || ⩾ 81.The remaining part of the proof is computational and has been checked by all three authors in the computer algebra systems GAP [4] and SageMath [13].
Thanks to Proposition 5.1, we know that  has maximal class.There exist precisely four groups of order 3 4 = 81 and maximal class up to isomorphism: these are the groups SmallGroup(81,7), SmallGroup(81,8), SmallGroup(81,9) and SmallGroup(81,10) in the SmallGroup library of GAP [3].Each of these groups has an automorphism  of order 2 that induces scalar multiplication by −1 on the Frattini quotient.For each of these groups other than SmallGroup(81,10), the subgroup generated by the elements of order 3 has order at least 27: this ensures that the group has a subgroup of order  2 on which  acts as scalar multiplication by −1, contradicting property (c).On the contrary, the subgroup of SmallGroup(81,10) that is generated by the elements of order 3 is equal to the derived subgroup of SmallGroup(81,10), from which it is not difficult to deduce that (SmallGroup(81,10), ) is a maximal pair of rank 2 yielding the 3-maximal group SmallGroup(162,22).
If we now move to the groups of order 3 5 = 243, we find that SmallGroup(243,26) is the unique 3-group, up to isomorphism, of maximal class and order 243 that possesses an automorphism  of order 2 that induces scalar multiplication by −1 on the Frattini quotient.However, the quotient of SmallGroup(243,26) by its centre is isomorphic to SmallGroup(81,9) and thus not isomorphic to SmallGroup(81,10).As a consequence of Lemma 2.9, we derive that SmallGroup(243,26) is not part of any maximal pair and our classification is therefore complete.□
(1) There exists an odd prime  such that  is a -group.Moreover,  is isomorphic to one of the following groups: (i) an elementary abelian group of order  3 ; (ii) the group of order  4 defined by (2) The group  is a 2-group.More precisely,  is isomorphic to one of the following groups: (i) an elementary abelian group of order 8; (ii) the direct product  2 ×  8 ; (iv) the central product  4 *  8 =  4 *  8 ; (iv) SmallGroup(32,32).(3) There exist an odd prime  and a positive integer  such that  is a semidirect product  ⋊ ⟨⟩ where  is a -group,  has order   for some prime  that divides  − 1,   ∈ Z() and ∕⟨  ⟩ is isomorphic to one of the following: (i) a semidirect product  ⋊   , where  is elementary abelian of order  2 ; (ii) a semidirect product  ⋊   with  extraspecial of exponent  and order  3 ; (iii) SmallGroup(162,22).
Define now  = ∕ 3 () and use the bar notation for the subgroups of .The automorphism  induces an automorphism  of  and it follows from Lemma 2.12 that  =  + ⊕  − . Let now  be a subgroup of  that contains  3 () and such that  =  − .Since  3 () =  3 () − , it follows from Lemma 2.13 that  =  − and  is abelian.Since  is contained in , this yields a contradiction to property (c) of maximal pairs.□ Proposition 6.3.Let (, ) be a maximal (, )-pair of rank 3.If  > 3 and || =  5 , then  = 2 and  is uniquely determined up to isomorphism.Indeed,  is isomorphic to where the following hold: Proof.The result could be deduced from the list of finite groups of order  5 , given by Bender in [2].In that paper, the groups of order   4 , then  is a maximal subgroup of , and therefore, Φ() = ⟨ 4 ,  5 ⟩ ⊆ .Moreover, either  = Ω 1 () or exp() =  2 .In any case,  5 belongs to Φ().So, d() ⩽ 3 and if d() = 3, then Φ() = ⟨ 5 ⟩.In the latter case, since  4 ∈  and ( 4 ) =  4 , the map  does not induce a non-trivial power automorphism of ∕Φ().Finally, suppose that  is elementary abelian of order  3 and that  induces a non-trivial power automorphism on .It must be that  is contained in Ω 1 () and  4 ∉ .This is impossible, because  would be a maximal subgroup of Ω 1 () and it would contain Φ(Ω 1 ()) = ⟨ 4 ,  5 ⟩.Proposition 6.5.Let (, ) be a maximal (, )-pair of rank 3.If  > 3, then the order of  is at most  5 .

A C K N O W L E D G E M E N T S
The second author has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No 741420).The third author was funded by the Italian program Rita Levi Montalcini for young researchers, Edition 2020.We are thankful to the anonymous referees for their comments, which helped improve the exposition of this paper.

J O U R N A L I N F O R M AT I O N
The Bulletin of the London Mathematical Society is wholly owned and managed by the London Mathematical Society, a not-for-profit Charity registered with the UK Charity Commission.All surplus income from its publishing programme is used to support mathematicians and mathematics research in the form of research grants, conference grants, prizes, initiatives for early career researchers and the promotion of mathematics.
It follows from Lemmas 3.1 and 3.2 that, if (, ) is a maximal (, )-pair of rank  and  has class at most , then || ⩽   ; the previous question is thus equivalent to the following one.Does there exist a function g ∶ ℕ → ℕ with the property that  has class at most g(), whenever (, ) is a maximal pair of rank ?We are not aware of examples of maximal pairs (, ) with  of class greater than 3.This motivates the next problem.
Theorem 2.1.Let  be a finite group.Let  = max ∈Syl() d(), where  runs among the Sylow subgroups of .Then, d() ⩽  + 1.If d() =  + 1, then there exists an odd prime  and a quotient of  isomorphic to a semidirect product of an elementary abelian -group  of rank  with a cyclic group ⟨⟩, where  acts on  as a non-trivial power automorphism.