The spread of ﬁnite and inﬁnite groups Scott Harper Abstract

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The spread of ﬁnite and inﬁnite groups

Scott Harper*

Abstract

It is well known that every ﬁnite simple group has a generating pair. Moreover,

Guralnick and Kantor proved that every ﬁnite simple group has the stronger

property, known as

-generation, that every nontrivial element is contained in a

generating pair. Much more recently, this result has been generalised in three

diﬀerent directions, which form the basis of this survey article. First, we look at

some stronger forms of

-generation that the ﬁnite simple groups satisfy, which

are described in terms of spread and uniform domination. Next, we discuss

the recent classiﬁcation of the ﬁnite

-generated groups. Finally, we turn our

attention to inﬁnite groups, focusing on the recent discovery that the ﬁnitely

presented simple groups of Thompson are also

-generated, as are many of

their generalisations. Throughout the article we pose open questions in this

area, and we highlight connections with other areas of group theory.

1 Introduction

Every ﬁnite simple group can be generated by two elements. This well-known

result was proved for most ﬁnite simple groups by Steinberg in 1962 [

] and

completed via the Classiﬁcation of Finite Simple Groups (see [

]). Much more

is now known about generating pairs for ﬁnite simple groups. For instance, for

any nonabelian ﬁnite simple group

, almost all pairs of elements generate

[

has an invariable generating pair [

], and, with only ﬁnitely

many exceptions,

can be generated by a pair of elements where one has order

2 and the other has order either 3 or 5 [80, 83].

*School of Mathematics and Statistics, University of St Andrews, St Andrews, KY16 9SS, UK

scott.harper@st-andrews.ac.uk

arXiv:2210.09635v2 [math.GR] 20 Apr 2023

The particular generation property of ﬁnite simple groups that this survey

focuses on was established by Guralnick and Kantor [

] and independently by

Stein [

]. They proved that if

is a ﬁnite simple group, then every nontrivial

element of

is contained in a generating pair. Groups with this property are

said to be

-generated. We will survey the recent work (mostly from the past

ﬁve years) that addresses natural questions arising from this theorem.

Section 2 focuses on ﬁnite groups and considers recent progress towards an-

swering two natural questions. Do ﬁnite simple groups satisfy stronger versions

-generation? Which other ﬁnite groups are

-generated? Regarding the

ﬁrst, in Sections 2.2 and 2.3, we will meet two strong versions of

-generation,

namely (uniform) spread and total/uniform domination. Regarding the second,

Section 2.4 presents the recent classiﬁcation of the ﬁnite

-generated groups

established by Burness, Guralnick and Harper in 2021 [

]. All these ideas are

brought together as we discuss the generating graph in Section 2.5. Section 2.6

rounds oﬀthe ﬁrst half by highlighting applications of spread to word maps,

the product replacement graph and the soluble radical of a group.

Section 3 focuses on inﬁnite groups and, in particular, whether any results on

the

-generation of ﬁnite groups extend to the realm of inﬁnite groups. After

discussing this in general terms in Sections 3.1 and 3.2, our focus shifts to the

ﬁnitely presented inﬁnite simple groups of Richard Thompson in Sections 3.3

to 3.6. Here we survey the ongoing work of Bleak, Donoven, Golan, Harper,

Hyde and Skipper, which reveals strong parallels between the

-generation of

these inﬁnite simple groups and the ﬁnite simple groups. Section 3.3 serves as

an introduction to Thompson’s groups for any reader unfamiliar with them.

This survey is based on my one-hour lecture at Groups St Andrews 2022

at the University of Newcastle, and I thank the organisers for the opportunity

to present at such an enjoyable and interesting conference. I have restricted

this survey to the subject of spread and have barely discussed other aspects of

generation. Even regarding the spread of ﬁnite simple groups, much more could

be said, especially regarding the methods involved in proving the results. Both

of these omissions from this survey are discussed amply in Burness’ survey

article from Groups St Andrews 2017 [

], which is one reason for deciding to

focus in this article on the progress made in the past ﬁve years.

Acknowledgements.

The author wrote this survey when he was ﬁrst a

Heilbronn Research Fellow and then a Leverhulme Early Career Fellow, and he

thanks the Heilbronn Institute for Mathematical Research and the Leverhulme

Trust. He thanks Tim Burness, Charles Cox, Bob Guralnick, Jeremy Rickard

and a referee for their helpful comments, and he also thanks Guralnick for his

input on Application 3, especially his suggested proof of Theorem 2.42.

2 Finite Groups

2.1 Generating pairs

It is easy to write down a pair of generators for each alternating group

: for

instance, if

is odd, then

(1 2 3)

(1 2

. . . n

)

. In 1962, Steinberg [

]

proved that every ﬁnite simple group of Lie type is 2-generated, by exhibiting

an explicit pair of generators. In light of the Classiﬁcation of Finite Simple

Groups, once the sporadic groups were all shown to be 2-generated, it became

known that every ﬁnite simple group is 2-generated [

]. Since then, numerous

stronger versions of this theorem have been proved (see Burness’ survey [

]).

Even as early as 1962, Steinberg raised the possibility of stronger versions of

his 2-generation result [95]:

“It is possible that one of the generators can be chosen of order 2, as is the case for the

projective unimodular group, or even that one of the generators can be chosen as an

arbitrary element other than the identity, as is the case for the alternating groups. Either

of these results, if true, would quite likely require methods much more detailed than

those used here.”

That is, Steinberg is suggesting the possibility that for a ﬁnite simple group

one might be able to replace just the existence of

x,y∈G

such that

hx,yi

with the stronger statement that for all nontrivial elements

x∈G

there exists

y∈G

such that

hx,yi

. He alludes to the fact that this much stronger

condition is known to hold for the alternating groups, which was shown by

Piccard in 1939 [

]. In the following example, we will prove this result on

alternating groups, but with diﬀerent methods than Piccard used.

Example 2.1.

Let

for

5. We will focus on the case

n≡

0 (

mod

and then address the remaining cases at the end.

Write

and let

have cycle shape [2

m−

+1], that is, let

be a

product of disjoint cycles of lengths 2

m−

1 and 2

+1. Visibly,

is contained

in a maximal subgroup

H6G

of type (

S2m−1×S2m+1

)

∩G

. We claim that no

further maximal subgroups of

contain

. Imprimitive maximal subgroups

are ruled out since 2

m−

1 and 2

+1 are coprime. In addition, a theorem

of Marggraf [

100

, Theorem 13.5] ensures that no proper primitive subgroup

contains a

-cycle for

k<n

, so

is contained in no primitive maximal

subgroups as a power of sis a (2m−1)-cycle.

Now let

be an arbitrary nontrivial element of

. Choosing

such that

moves some point from the (2

m−

1)-cycle of

to a point in the (2

+1)-cycle

gives

x<Hg

. This means that no maximal subgroup of

contains both

and

, so

hx,sgi

. In particular, every nontrivial element of

is contained

in a generating pair.

We now address the other cases, but we assume that

25 for exposition.

n≡

2 (

mod

4), then we choose

with cycle shape [2

m−

+3] (where

+2) and proceed as above but now the unique maximal overgroup has

type (

S2m−1×S2m+3

)

∩G

. A similar argument works for odd

. Here

has cycle

shape [

m−

,m,m

+2] if

, [

,m−

1] if

+1 and

[

,m,m

] if

+2, and the only maximal overgroups of

are the three

obvious intransitive ones. For each 1

,x∈G

, it is easy to ﬁnd

g∈G

such that

misses all three maximal overgroups of

and hence deduce that

hx,sgi

In 2000, Guralnick and Kantor [

] gave a positive answer to the longstanding

question of Steinberg by proving the following.

Theorem 2.2.

Let

be a ﬁnite simple group. Then every nontrivial element of

G is contained in a generating pair.

We say that a group

-generated if every nontrivial element of

contained in a generating pair. The author does not know the origin of this

term, but it indicates that the class of

-generated groups includes the class of

1-generated groups and is included in the class of 2-generated groups. This is

somewhat analogous to the class of

-transitive permutation groups introduced

by Wielandt [

100

, Section 10], which is included in the class of 1-transitive

groups and includes the class of 2-transitive groups.

Let us ﬁnish this section by brieﬂy turning from simple groups to simple Lie

algebras. Here we have a theorem of Ionescu [72], analogous to Theorem 2.2.

Theorem 2.3.

Let

be a ﬁnite dimensional simple Lie algebra over

. Then

for all

x∈g\

0there exists

y∈g

such that

and

generate

as a Lie algebra.

In fact, Bois [

] proved that every classical ﬁnite dimensional simple Lie

algebra in characteristic other than 2 or 3 has this

-generation property, but

Goldstein and Guralnick [56] have proved that slnin characteristic 2 does not.

2.2 Spread

Let us now introduce the concept that gives this article its name.

Deﬁnition 2.4. Let Gbe a group.

(i)

The spread of

, written

(

), is the supremum over integers

such that

for any

nontrivial elements

x1,...,xk∈G

there exists

y∈G

such that

hx1,yi=··· =hxk,yi=G.

(ii)

The uniform spread of

, written

(

), is the supremum over integers

for which there exists

s∈G

such that for any

nontrivial elements

x1,...,xk∈Gthere exists y∈sGsuch that hx1,yi=··· =hxk,yi=G.

The term spread was introduced by Brenner and Wiegold in 1975 [

], but

the term uniform spread was not formally introduced until 2008 [16].

Note that

(

)

0 if and only if every nontrivial element of

is contained in

a generating pair. Therefore, spread gives a way of quantifying how strongly a

group is

-generated. Uniform spread captures the idea that the complementary

element

, while depending on the elements

x1,...,xk

, can be chosen somewhat

uniformly for all choices of

x1,...,xk

: it can always be chosen from the same

prescribed conjugacy class. In Section 2.3, we will see a way of measuring

how much more uniformity in the choice of

we can insist on. Observe that

Example 2.1 actually shows that u(An)>1 for all n>5.

By Theorem 2.2, every ﬁnite simple group Gsatisﬁes s(G)>0. What more

can be said about the (uniform) spread of ﬁnite simple groups? The main result

is the following proved by Breuer, Guralnick and Kantor [16].

Theorem 2.5.

Let

be a nonabelian ﬁnite simple group. Then

(

)

(

)

Moreover, s(G)=2if and only if u(G)=2if and only if

G∈ {A5,A6,Ω+

8(2)}∪{Sp2m(2) |m>3}.

The asymptotic behaviour of (uniform) spread is given by the following

theorem of Guralnick and Shalev [

, Theorem 1.1]. The version of this

theorem stated in [

] is given just in terms of spread, but the result given here

follows immediately from their proof (see [65, Lemma 2.1–Corollary 2.3]).

Theorem 2.6.

Let (

)be a sequence of nonabelian ﬁnite simple groups such

that

|Gi| → ∞

. Then

(

)

→ ∞

if and only if

(

)

→ ∞

if and only if (

)

has no inﬁnite subsequence consisting of either

(i) alternating groups of degree all divisible by a ﬁxed prime

(ii)

symplectic groups over a ﬁeld of ﬁxed even size or odd-dimensional or-

thogonal groups over a ﬁeld of ﬁxed odd size.

Given that

(

)

→ ∞

if and only if

(

)

→ ∞

, we ask the following. (Note

that s(G)−u(G) can be arbitrarily large, see Theorem 2.9(iv) for example.)

Question 2.7.

Does there exist a constant

such that for all nonabelian ﬁnite

simple groups Gwe have s(G)6c·u(G)?

There are explicit upper bounds that justify the exceptions in parts (i) and (ii)

of Theorem 2.6. Indeed,

(

Sp2m

(

))

for even

and

(Ω

2m+1

(

))

(

)

for odd

(see [

, Proposition 2.5] for a geometric proof). For alternating

groups of composite degree

4, if

is the least prime divisor of

, then

(

)

62p+1

3

(see [

, Proposition 2.4] for a combinatorial proof). For even-

degree alternating groups, the situation is clear:

(

)=4, but much less is

known in odd degrees (see [65, Section 3.1] for partial results).

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ThespreadofniteandinnitegroupsScottHarper*AbstractItiswellknownthateverynitesimplegrouphasageneratingpair.Moreover,GuralnickandKantorprovedthateverynitesimplegrouphasthestrongerproperty,knownas32-generation,thateverynontrivialelementiscontainedinageneratingpair.Muchmorerecently,thisresulthasbeen...

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