Quasipowerful groups
Abstract.
In this paper we introduce the notion of a quasipowerful group for odd primes . These are the finite groups such that is powerful in the sense of Lubotzky and Mann. We show that this large family of groups shares many of the same properties as powerful groups. For example, we show that they have a regular power structure, and we generalise a result of FernándezAlcober on the order of commutators in powerful groups to this larger family of groups. We also obtain a bound on the number of generators of a subgroup of a quasipowerful group, expressed in terms of the number of generators of the group. We give an infinite family of examples which demonstrates this bound is close to best possible.
2010 Mathematics Subject Classification:
Primary: 20D15, Secondary: 20D1. Introduction
It can be said that the modern study of finite groups began with the groundbreaking paper of P. Hall, published in 1933 [8]. In this paper Hall introduced the notion of a regular group (see Definition 2.13) and he showed that these groups have desirable properties and a theory which in some sense parallels that of abelian groups. The study of families of finite groups with certain desirable properties continues to this day.
The powerful groups, introduced by Lubotzky and Mann in [12], are another wellstudied family of groups with abelianlike properties. Powerful groups also have many abelianlike properties. On the one hand they can be thought of as very similar to abelian groups, but on the other hand they can be thought of as close to a typical group (see Remark 2.10). Given this, it is not surprising that the theory of powerful groups has found many applications, as problems concerning typical groups can often be reduced to questions about powerful groups. Perhaps the most widely celebrated application is Shalev’s proof of the coclass conjectures [18]. However the impact of powerful groups is extensive and even stretches beyond finite groups (for more applications, see Remark 8.1).
Given the widespread usefulness of powerful groups, it is natural to seek a larger family of groups with similar properties. With this goal in mind, we introduce quasipowerful groups, and we extend many of the remarkable properties of powerful groups to this family.
Definition.
Let be an odd prime. We say that a finite group is quasipowerful if is a powerful group.
This family is larger than that of powerful groups. For instance it contains all powerful groups and also all groups of nilpotency class , and therefore unifies two large families of groups which are known to have abelianlike properties. Informally we can think of groups in this family as being close to powerful groups; in a powerful group any commutator is equal to a th power in the group, whereas in a quasipowerful group any commutator is equal to the product of a th power and an element in the centre. It seems reasonable to expect that many properties of powerful groups can be adapted to this family of groups and in this paper we show this to be the case. It is our hope that quasipowerful groups will be useful as a tool in inductive arguments and reductions.
In order to state our main results, we need to recall some notation and terminology. Let be a finite group. The th Omega subgroup of is defined by
Notice that this coincides with the set of elements whose order divides if and only if for any two elements each of order at most we have that the order of the product is at most . Another equivalent formulation of this is that . This is clearly true for abelian groups, however this is not true for groups in general. For example, contains an element of order .
The th Agemo subgroup of is defined by
Sometimes this subgroup is denoted as . Note that the set of th powers need not coincide with the group it generates. However, in the abelian setting they do coincide.
As the terminology and notation indicates, the Omega and Agemo subgroups are in some sense dual to each other. Indeed, for an abelian group we have that for all integers .
These ideas motivate the notion of a regular power structure.
Definition.
A finite group has a regular power structure if the following three conditions hold for all positive integers :
(1.1)  
(1.2)  
(1.3) 
We have seen above that abelian groups have a regular power structure. P. Hall showed in [8] that regular groups have a regular power structure. For odd primes , powerful groups also have a regular power structure  the first property (1.1) was established in [12], with the latter two properties (1.2) and (1.3) proved by Wilson in [24] (using a result of Héthelyi and Lévai [9]). An alternate and shorter proof of these last two facts is given by FernándezAlcober in [4]. Another independent proof that (1.3) holds in powerful groups is given by Mazur in [15].
In [6] GonzálezSánchez and JaikinZapirain introduce the family of potent groups (we recall this notion in Definition 2.11), and they show that this family of groups has a regular power structure (this property is called power abelian in [6]). Hence we see there is a sustained and significant interest not only in finding families of finite groups with a regular power structure, but also in exploring different approaches to the problem.
Our first main result reveals that quasipowerful groups have a regular power structure.
Theorem 1.
Let be an odd prime, let be a quasipowerful group and let be a nonnegative integer.

If both have order at most then their product has order at most .

The group generated by the th powers coincides with the set of th powers.

.
In particular, has a regular power structure.
We will show that if , then every quasipowerful group is potent; thus for the fact they have a regular power structure follows from [6]. However all results given in this paper will be proved independently of this fact and moreover for us the difficult and most interesting case is and the quasipowerful groups need not be potent (see Example 3.2).
It is well known that if is a powerful group then is also powerful. In fact, this is true for all known families of groups satisfying property (1.1). It is an open question if the same condition holds for any finite group satisfying (1.1). This question was posed by Wilson in [24] and he noted that an affirmative answer to this question would answer a question of Shalev [19, Problem 13] (see Question 4.8). Since quasipowerful groups satisfy (1.1), Wilson’s question provides additional motivation for our next result.
Theorem 2.
Let be a quasipowerful group. Then is a powerful group for all .
To understand finite groups, one needs to study the interactions between powers and commutators. For example, the defining properties of all of the families of groups described above all involve some condition on commutators in terms of groups of th powers. Theorem 1 in [4] gives bounds on the order of a commutator in a powerful group in terms of the components within the commutator (see Theorem 2.6). These results turn out to be very useful when working with th powers in powerful groups, and played a key role in [22] and [23] to show that certain normal subgroups of powerful groups are powerfully nilpotent (see Section 8.4).
We generalise these bounds on commutators to quasipowerful groups. The first result shows that in a quasipowerful group the order of a commutator is bounded by the order of its components.
Theorem 3.
Let be a quasipowerful group and . Then .
The next result allows us to say more if we know how elements can be expressed as th powers.
Theorem 4.
Let be a quasipowerful group. If are such that and , then for all
An interesting point to note is that we apply Theorem 1(i) to prove Theorem 4. The argument we use is quite general and can be applied to prove an analagous result for potent groups for odd primes , hence we can obtain the following theorem.
Theorem 5.
Let be an odd prime and be a potent group. Suppose with and , then for all .
One of the most abelianlike and important properties of powerful groups is that the minimal number of generators of a subgroup is bounded by the minimal number of generators of the group. We prove an extension of this result for quasipowerful groups, and we show that the given bound is close to best possible. Let denote the minimum number of generators for .
Theorem 6.
Let be a quasipowerful group with and let . Then .
Furthermore, we exhibit an infinite family of examples of groups such that but with a subgroup such that . Hence the bound in Theorem 6 is close to best possible.
We now say a few words on the layout of this paper. First, in Section 2 we recall some preliminary results and definitions from the theory of finite groups. Next, in Section 3 we introduce some basic properties of quasipowerful groups. We include an example to demonstrate that quasipowerful groups are a ‘new’ family with regular power structure. The section culminates with the proof of Theorem 3. Section 4 is split into three parts, with each part corresponding to establishing a part of Theorem 1. In Section 5 we give an application of Theorem 1, whereby we prove Theorem 4. Our argument can be adapted to potent groups (as in Theorem 5). Our focus turns to the minimal number of generators of subgroups in Section 6. In Section 7 we comment on the case when . We give an example to show that if the definition of a quasipowerful group for odd primes were extended to , the groups need not have a regular power structure.
Notation: Our notation is standard. We denote the order of as . In keeping with [4], we use the convention that if is a group and , then we define the meaning of the inequality with to be that . We denote the exponent of by . All iterated commutators are left normed. The terms of the lower central series of are defined recursively as and for integers . We use bar notation for images in a quotient group; it will always be made explicitly clear what the quotient group under consideration is. We denote the minimal number of generators of a group by . We denote the Frattini subgroup of by .
Acknowledgements
I would like to thank Dr Tim Burness for many helpful discussions and for his detailed feedback on earlier versions of this paper. I am also very grateful to Dr Gareth Tracey for his continued encouragement with this project.
2. Preliminaries
In all of what follows we shall be dealing with finite groups where is an odd prime unless explicitly stated otherwise. For the convenience of the reader, we collect here some properties of groups which shall be used in the rest of the paper. Most of the results are standard, but we draw the reader’s attention to Theorem 2.6 and Theorem 2.12 which have appeared relatively recently in the literature.
2.1. Powerful groups
We recall from [12] what it means for a group to be powerfully embedded and for a group to be powerful.
Definition 2.1.
A subgroup of a finite group is powerfully embedded in if (for , if ). A finite group is powerful if it is powerfully embedded in itself, that is, if (for , if ).
The following theorem demonstrates why powerful groups are so named  because they are full of th powers. Theorem 1(ii) and Theorem 2 generalise this theorem.
Theorem 2.2 ([11], Theorem 11.10).
Let be a powerful group, and let .

The subgroup coincides with the set of th powers of elements of ; in particular, for all .

is powerfully embedded in .
Lemma 2.3 ([11], Lemma 11.2).
A normal subgroup in a finite group is powerfully embedded in if .
Lemma 2.3 will be used in this paper when we wish to show that certain subgroups are powerfully embedded as it allows for a reduction to a simpler case by assuming .
Lemma 2.4 ([11], Lemma 11.7).
If is a powerfully embedded subgroup of , then, for any , the subgroup is a powerful group and .
Lemma 2.4 will enable us to reduce problems about quasipowerful groups to powerful groups. We will show in Proposition 3.6 that for a quasipowerful group , the subgroup is powerfully embedded in . Hence for any we will know that is a powerful group.
Lemma 2.5 (Interchanging Lemma, [17], Lemma 3.1).
If and are powerfully embedded subgroups in a finite group , then for all .
Generalising the next theorem to the case when is a quasipowerful group is one of the main aims of this paper. We remark that we are sometimes able to deploy this theorem directly by reducing from a quasipowerful group to a powerful subgroup.
Theorem 2.6 ([4], Theorem 1).
Let be a powerful group. Then, for every :

If and then .

If are such that and , then for all .

If is odd, then .
The following result was originally proved in [24], but an alternate proof was given in [4] and [15].
We now list two of the most abelianlike properties of powerful groups with respect to generators and subgroups.
Theorem 2.8 ([12], Theorem 1.12).
Let be a powerful group with and let . Then .
Theorem 2.9 ([12], Theorem 1.11).
Let be a powerful group with , then is a product of cyclic groups.
We obtain variants of both of these results for quasipowerful groups in Section 6.
Remark 2.10.
The properties of powerful groups given so for demonstrate many of the abelianlike features of powerful groups. Thus on the one hand we can think of powerful groups as being close to abelian groups. On the other hand it turns out that we can think of powerful groups as being close to a typical group. For example, by a result of Lubotzky it is known that every finite group appears as a section of a powerful group (see [14], Theorem 1). Additionally it is known that if all characteristic subgroups of a finite group can be generated by elements, then contains an generator, powerful, characteristic subgroup whose index is bounded in terms of and ([12], Theorem 1.14).
We close this discussion on powerful groups by recalling a result from [6]. First we need the following definition.
Definition 2.11.
A finite group is potent if for or for .
Note that for , the definitions of potent and powerful coincide.
Theorem 2.12 ([6], Theorem 5.1).
Let be a powerful group and a normal subgroup of . Then one of the following two properties holds:

For any such that if is odd, .

There exists a proper powerful subgroup of such that .
2.2. Regular groups
Regular groups were introduced by P. Hall in his pioneering paper [8].
Definition 2.13.
Let be a finite group. We say is a regular group if for every we have that for some .
We now recall two results from the theory of regular groups. The first gives a condition on when a group is regular, based on the nilpotency class of the group. The second result tells us that in a regular group the order of a product of two elements cannot exceed the order of the factors.
Theorem 2.14 ( [8], Corollary 4.14, Theorem 4.26).
Let be a finite group.

If the nilpotency class of is less than then is regular.

If and are any two elements of the regular group , then the order of cannot exceed the orders of both and . In particular for any the subgroup
3. Basic Properties of Quasipowerful groups
In this section we introduce the basic properties of quasipowerful groups for odd primes . The results proved in this section will be used throughout the rest of the paper. Proposition 3.6 will sometimes allow us to reduce to a powerful subgroup within a quasipowerful group. At the end of this section this idea is used to prove Theorem 3.
Definition 3.1.
Let be an odd prime, we say that a group is a quasipowerful group if is a powerful group.
We do not give a definition of quasipowerful groups for , but in Section 7 we give an example which suggests for a different definition is needed.
Suppose that is powerful. Throughout the rest of this paper, we shall let . Notice that . From this it is clear that powerful groups and groups of nilpotency class are quasipowerful groups. However there exist quasipowerful groups which are neither of those things. We now give an example of a group of nilpotency class such that is powerful, but is neither regular nor powerful.
Example 3.2.
Let be the group with presentation:
The following details are easily checked using GAP [5], where this group can be constructed as SmallGroup(6561,86718) using the package SglPPow [21]. We can describe the structure of this group as
In addition, we have and . The group is not powerful because . However is powerful, since . Furthermore since for the definitions of potent and powerful coincide, is also not a potent group.
Moreover one can show that this group is not a regular group. For example let and . Then
Example 3.2 demonstrates a quasipowerful group which does not fall into one of the families which are already known to have a regular power structure.
Before moving on we make the following remark which will be used frequently throughout the rest of this paper.
Remark 3.3.
It is easy to see that the property of being a quasipowerful group is preserved under taking quotients. However it is not necessarily preserved under taking subgroups. For instance in Example 3.2, one can check that the subgroup is not quasipowerful.
We now begin by investigating the subgroup .
Lemma 3.4.
For any we have that for some and .
Proof.
Since is powerful, in the quotient group the product of th powers is a th power and so for some . Thus for some and . ∎
Remark 3.5.
This means that any is of the form for some and , since , so by repeated application of Lemma 3.4.
Proposition 3.6.
The subgroup is powerfully embedded in .
Proof.
By Lemma 2.3, we may assume that . Consider some . By Remark 3.5 we can write for some and . Let and consider
(3.1) 
We have of the terms and of the terms. Observe that . Hence these terms of weight are central because . Also notice that this implies . Hence (3.1) becomes
As and and is an odd prime we see that . It follows that . ∎
Remark 3.7.
As is powerfully embedded in , we can conclude that for the group is potent, since
Thus if we could appeal to [6] to conclude that these groups have a regular power structure. However, we will give an independent proof of this fact and deal with all odd primes. Furthermore we will see the most involved and interesting case is when , and we have already seen that a quasipowerful group need not be potent.
Proof of Theorem 3.
Remark 3.8.
In fact this property is true for any group containing a powerfully embedded subgroup with .
Theorem 3 will be used frequently throughout the remainder of the paper.
4. Quasipowerful groups have regular power structure
4.1. The exponent of omega subgroups
We will see that by using properties of powerful groups and regular groups we can relatively quickly deal with the case of primes greater than . However when we can no longer assume the subgroup is regular and the situation becomes more involved.
Lemma 4.1.
If , where are elements of order , then
Proof.
Lemma 4.2.
The nilpotency class of is at most .
Proof.
We note that is generated by the elements in of order , say . By Lemma 4.1 it is clear that any commutator of weight in these must be trivial. ∎
Proposition 4.3.
If is a prime such that , then has exponent at most .
Proof.
We now begin to deal with the difficult case of . We will eventually see that we can reduce to the case where is a quasipowerful group with cyclic centre and .
Lemma 4.4.
Let be a finite group of nilpotency class at most such that is powerful. Then .
Proof.
Let both be of order . We expand , making use of the fact that the nilpotency class is at most .
By Theorem 3 we know that any commutator containing or has order at most . Hence we obtain:
As is powerful, we can write for some and . Then
where again we use that the nilpotency class is at most and the fact that any commutator containing the element has order at most . That the commutator is trivial follows in a similar way. Thus we can conclude that if and have order at most then so does and hence . ∎
Next we explain how we can make the reduction to the case where the centre of our quasipowerful group is cyclic and that in our goal to show that .
Suppose that a group is a quasipowerful group of smallest order such that the exponent of is greater than . In this case there must exist elements and in both of order such that . Let be a subgroup of order . Then is a quasipowerful group of smaller order, and so in this group we must have that has order at most , in other words . This allows us to assume that . Furthermore, if the centre of is not cyclic then it would contain two distinct subgroups and of order such that . Then we would be able to conclude that were in both, and thus . Thus for what follows we consider a quasipowerful group of minimal order such that and therefore we may assume the centre is cyclic and that .
Proposition 4.5.
Let be a quasipowerful group. Then the exponent of is at most .
Proof.
By the discussion above, we can assume that has cyclic centre and that .
Consider the subgroup . Notice that is normal (in fact characteristic) in . We use bar notation to denote images under the natural map corresponding to quotienting by . Consider the image, of in . The subgroup is normal in the powerful group . Then by Theorem 2.12 we have two cases to consider.
In the first case, we can assume that we have . Notice that , since . Hence . Now, using the Interchanging Lemma 2.5, since is powerful, we have that
Hence has class at most , and so has class at most . Then by Lemma 4.4 we obtain the desired result.
In the second case we can assume that is contained in some proper powerful subgroup of . Call this subgroup . Then lifting up to the proper subgroup of where , we see that
Hence is a quasipowerful group of order strictly less than . Therefore the exponent of is at most . In particular, the product of two elements of order in has order at most . Then since we conclude that the exponent of is at most . ∎
Thus we see that for any odd prime we have that . This now enables us to prove the more general result that for any and any odd prime  this is Theorem 1(i).
Proof of Theorem 1 (i).
Let be a quasipowerful group and . We proceed by induction on the order of . The result is clearly true for groups of order . Now suppose that and that the claim holds for all quasipowerful groups of smaller order. Notice that if with having order then and thus has order at most in . It follows that
Then by the inductive hypothesis, since is a quasipowerful group of smaller order, we have that the exponent of is at most and consequently the subgroup has exponent at most . Then lifting back up to we see that and by Proposition 4.3 and Proposition 4.5 it follows that . ∎
4.2. th powers
In this section we prove properties about the groups of th powers in quasipowerful groups. We will prove Theorem 1 (ii) and Theorem 2.
We first prove that just like in a powerful group, the product of th powers in a quasipowerful group is equal to a th power. This is the first step in proving Theorem ii.
We will need to recall the following formulation of the collection formula of Philip Hall (see [16], Exercise 1.2). If is a group, and then
(4.1) 
where .
Theorem 4.6.
Let be a quasipowerful group and an odd prime. Then
Proof.
Let . We recall that . Using the collection formula (4.1) we have
where . Now notice that and so . Also notice that since we have that because is powerfully embedded by Proposition 3.6. Now since is powerful by Proposition 3.6, contains precisely the th powers of elements of . Thus we have that
for some . Then . Now let . By Lemma 2.4 and Proposition 3.6, it follows that is powerful and so
for some . Thus . ∎
Question 4.7.
If is a group (with odd) and the th powers of elements of form a subgroup, must this subgroup be powerful?
Wilson argues how an affirmative answer to this question would in turn provide an affirmative answer to a question of A. Shalev [19, Problem 13].
Question 4.8.
Let be a finitely generated pro group, and suppose that for each there is such that ; does it follow that is adic analytic?
This provides some motivation for our next results, as in light of this question it is natural to ask whether must be powerful for quasipowerful groups. We will need the following version of Philip Hall’s commutator expansion formula (see [16], Exercise 1.2). Let be a group, and , then
(4.2) 
where
Theorem 4.9.
Let be an odd prime. If is a quasipowerful group then is powerful.
Proof.
We wish to show that . By Lemma 2.3 we may assume that . Also as we can quotient by and in particular can assume that has exponent at most . Consider . We will show that . Using the collection formula (4.2) we have:
where . We need to show that and . First we show that .
Recall that by Theorem 1(i), the product of elements of order has order at most , and therefore we only need to show that the generators of have order at most to be able to conclude that all elements have order at most . Notice that for some and . As the exponent of is at most we can assume that has order at most , and then by Theorem 3 it follows that any commutator which includes the element as a term, has order at most . Therefore it follows that every element in has order at most and so .
Next we show that . Observe that because is powerful and of exponent at most we must have that . Using again the fact that we can write and that we have that .
Thus we have that and since is normal and so as required. ∎
Remark 4.10.
Although is powerful, may not be powerfully embedded in (see Remark 8.3).
4.3. The index of agemo subgroups
We now move on to proving the final condition to show that quasipowerful groups have a regular power structure. Our arguments in this section rely heavily on the results of L. Wilson in [24].
Recall that a group is said to have a regular power structure if the following three conditions hold for all positive integers :
The first and second conditions have been established. Thus all that remains is to prove the final condition, that . To begin we prove a base case, when .
Proposition 4.11.
Let be a quasipowerful group, then .
Proof.
Suppose the result holds for all quasipowerful groups of smaller order. We consider two cases depending on the exponent of . For the first case, suppose that has exponent . Here we can assume there must be an element of order with otherwise would be powerful and we are done. Let and . Suppose . Notice first that since We need to consider We have . If but it would mean that but then , a contradiction. Thus we must have . Then , that is
and the result follows.
Now for the second case, suppose that the exponent of is greater than . In this case we can find some element of order such that is central in . Let . Notice that
Thus if we can show that then we are done. Observe that is of order and so . Consider the cosets of in , where . Then we can assume that , and so . Next observe that is of order in and so but , the image of in , has order and so . Since is central, it is easy to see that has order . We now show that everything in is accounted for. Suppose that but . Then for some , but then and so but then . Hence we can conclude that and the result follows. ∎
We now prove the more general result by induction. We remark that our proof below follows Wilson’s proof of Theorem 3.1 in [24] very closely. Moreover, we will call upon the following result of Wilson taken from [24]. We will let be the class of all groups for which is the set of elements of order dividing for all .
Lemma 4.12 ([24], Lemma 2.1).
Let be in . Then for all and
Proof of Theorem 1(iii).
We use induction on . We have established the base case in Proposition 4.11. Assume now that the result holds for . We wish to find the order of . By Theorem 1 (ii) we have that As is powerful, we can apply Theorem 2.7 and conclude that
(4.3) 
Now since quotients of quasipowerful groups are quasipowerful