Automorphisms of Quotients of Symmetric Groups

J. L. Alperin

Department of Mathematics University of Chicago 5734 University Avenue Chicago Illinois 60637 U.S.A. ^{∗}

Jacinta Covington

Department of Mathematics Royal Melbourne Institute of Technology GPO Box 2476V Melbourne, Victoria 3001 Australia ^{†}

Dugald Macpherson

Department of Pure Mathematics University of Leeds Leeds LS2 9JT England

Abstract We describe the automorphism group of the quotient of the symmetric group on a countably infinite set by the group of finitary permutations. We show that the outer automorphism group is infinite cyclic. In the proof, we examine the structure of centralizers in this quotient.

1 Introduction

In this paper, S will denote Sym(Ω), the full symmetric group on a countably infinite set Ω. For any g ∈ S, define the support of g to be supp(g) = {α ∈ Ω: αg ≠ α} and the fixed point set to be fix g = {α ∈ Ω: αg = α}. It is well-known (see Scott [2] or originally Baer [1]) that S has precisely two proper non-trivial normal subgroups. These are F S(Ω) := {g ∈ S: | supp(g)| < ℵ_{0}} and Alt(Ω), the subgroup of index two in F S(Ω) consisting of even permutations. Throughout this paper, we let G := S/F S(Ω). This group is simple. The main goal of this paper is to describe Aut(G).

∗This research was partially supported by a grant from the National Science Foundation.

†This research was partially supported by ARC grant A68830987.

Theorem 1.1 Let G = Sym(Ω)/F S(Ω), where Ω is a countably infinite set. Then the outer automorphism group Out(G) is infinite cyclic.

To see why G admits an infinite cyclic group of outer automorphisms, observe that if Ω is identified with N, then the shift n ↦ n + 1, acting on G ‘by conjugation up to finitely many mistakes’, induces an outer automorphism of infinite order.

We give in Section 2 a more formal description of the infinite cyclic group of outer automorphisms. In Section 3 we describe in Proposition 3.2 and Theorem 3.3 the structure of centralizers in G, and thereby show that certain conjugacy classes of G are invariant under Aut(G). We believe that these results are of independent interest, as the centralizers do not have as simple a structure as might first be imagined. In Section 4, we show that there are no outer automorphisms other than those described in Section 2, and thereby prove Theorem 1.1.

We frequently refer to elements of S by their cycle type. The set of cycle lengths is the set of cardinals {n: 1 ≤ n ≤ ω}, and a nontrivial cycle has length n > 1. If T is a set of cycle lengths, we say that g has cycle type ∏(n^{a_{n}}: n ∈ T) if g has exactly a_{n} cycles of length n for each n ∈ T, (0 ≤ a_{n} ≤ ω), and no others. Observe that if g, h ∈ S have cycle types 1^{ω}. ∏(n^{a_{n}}: 1 < n ≤ ω) and 1^{ω}. ∏(n^{b_{n}}: 1 < n ≤ ω) respectively, then F S(Ω)g, F S(Ω)h are conjugate in G if and only if their cycle types differ finitely: that is, if

(a) a_{n} = b_{n} for all but finitely many finite cycle lengths n;

(b) a_{ω} = b_{ω};

(c) a_{n} = ω if and only if b_{n} = ω, for 1 ≤ n < ω.

It is important that here g and h have infinitely many fixed points. As a small warning, we remark that G has three conjugacy classes of involutions, corresponding to elements in S of the cycle types 1^{ω} · 2^{ω} and 2^{ω} and 1^{1} · 2^{ω}.

If g ∈ S then ḡ will be the corresponding element F S(Ω)g of G, and similarly, given ḡ ∈ G, we use g to denote any preimage of it under the natural surjection from S to G. We shall use ^{△} to denote symmetric difference, and write U ≡ V if U^{△}V is finite. Also, we extend the notion of support to elements ḡ ∈ G. If g ∈ S has support Γ ⊆ Ω, then ḡ has support

supp(ḡ) = {Δ ⊆ Ω: |Γ^{△}Δ| < ω}.

Thus supp(ḡ) lies in the Boolean Algebra B = Pow(Ω)/I, where I is the ideal of finite subsets of Ω. Observe the distinction between supp(g), which is a subset of Ω, and supp(ḡ), which is an element of B. All wreath products whose top group is a permutation group of infinite degree will be unrestricted, and similarly all direct products will be unrestricted (that is, will be the full Cartesian product). Finally, recall that a moiety Γ of a set Ω is a subset satisfying |Γ| = |Ω \ Γ|; for a countably infinite set Ω this is an infinite coinfinite subset.

The paper by J.K. Truss in the present volume sketches an alternative proof of Theorem 1.1, due to Truss and Shelah, in which the Boolean algebra B is shown to be first-order interpretable in G. Their approach seems to be more powerful,

since they obtain some generalisations of Theorem 1.1 for symmetric groups of uncountable cardinality. The strongest results concerning the interpretability of B in G were obtained by M. Rubin in Theorem 4.2 of [8]. There, he showed the following. Let μ < κ ≤ λ^{+} be infinite cardinals and G_{λκμ} denote the group of permutations (of support of size less than κ) of λ, modulo the normal subgroup of permutations of support of size less than μ, and B_{λκμ} be the corresponding Boolean algebra. Then B_{λκμ}, its boolean algebra structure, and the action of G_{λκμ} on it, is firstorder interpretable in the abstract group G_{λκμ}, and furthermore this interpretation is uniform, that is, the formulas in the interpretation are independent of the choice of λ, κ, μ. It seems, though, that the step from such results to a description of Aut(()G_{λκμ} is never immediate. Our approach yields en route the group-theoretic information about centralizers.

Acknowledgement. We would like to thank both Peter Neumann and John Truss for a number of very helpful remarks.

2 The group of near symmetries

In this section we describe an infinite cyclic group of outer automorphisms of G = Sym(Ω)/F S(Ω), where Ω is a countably infinite set. In the rest of the paper, we will show that this group is the whole outer automorphism group.

Define the set of near bijections of Ω to be

NB(Ω) = {f: f is a bijection Ω_{1} → Ω_{2}, where Ω_{1}, Ω_{2} are cofinite subsets of Ω}.

Clearly, S ⊆ NB(Ω). Extending our notation for sets, we define an equivalence relation ≡ on NB(Ω) by the rule

f_{1} ≡ f_{2} if there is a cofinite Ω′ ⊆ Ω such that f_{1} |_{Ω′}= f_{2} |_{Ω′} .

It is easily seen that ≡ is indeed an equivalence relation. We define NS(Ω), the set of near symmetries of Ω, to be the set of equivalence classes of this relation. This set naturally forms a group induced from composition of maps.

The natural projection from S to NS(Ω) is a group homomorphism with kernel F S(Ω). Hence we have an exact sequence

F S(Ω) ⟶ S ⟶ NS(Ω).

Thus G ≤ NS(Ω). The elements of G are the equivalence classes of permutations. For f ∈ NB(Ω) with Ω_{1}, Ω_{2} as above, we define the index of f to be

ind f = |Ω − Ω_{2}| − |Ω − Ω_{1}|.

Lemma 2.1 The map ind : NB(Ω) → Z induces a mapping ind : NS(Ω) → Z which is a homomorphism onto (Z, +) with kernel G.

Proof. First note that if Ω′ is a cofinite subset of Ω then for any f ∈ NB(Ω) we have ind f = ind (f |_{Ω′}). Take f, g ∈ NB(Ω) with f ≡ g. We want to show that ind f = ind g. There is a cofinite subset Ω′ of Ω such that f |_{Ω′}= g |_{Ω′}. But then clearly

ind f = ind (f |_{Ω′}) = ind (g |_{Ω′}) = ind g.

Thus ind is a well-defined function on NS(Ω).

We now show that ind (fg) = ind f + ind g. Restricting the domains of f and g if necessary, we may assume that dom g is equal to ran f, the range of f, that is, we have f: Ω_{1} → Ω_{2} and g: Ω_{2} → Ω_{3}. Then

ind (fg) = |Ω − Ω_{3}| − |Ω − Ω_{1}|
= |Ω − Ω_{3}| − |Ω − Ω_{2}| + |Ω − Ω_{2}| − |Ω − Ω_{1}|
= ind g + ind f.

Now the kernel of this homomorphism consists of all bijections f: Ω_{1} → Ω_{2} such that |Ω − Ω_{1}| = |Ω − Ω_{2}|. Any such map is ≡-equivalent to a permutation. Thus ker(ind ) = G.

From this lemma it follows that G is a normal subgroup of NS(Ω) and we have an exact sequence

1 ⟶ F S(Ω) ⟶ S ⟶ NS(Ω) ⟶ Z ⟶ 1.

Lemma 2.2 C_{NS(Ω)}(G) is trivial.

Proof. Let h be a near-bijection representing an element of C_{NS(Ω)}(G). Then for every permutation g ∈ S, we have h^{−1}gh ≡ g. Now, for any g ∈ S, the sets supp(h^{−1}gh) and (supp(g))h have finite symmetric difference, where supp(h^{−1}gh) is the the set of α ∈ Ω such that αh^{−1}gh is defined and not equal to α. It follows that supp(g) ≡ supp(g)h for any g ∈ S. Since any set Φ ⊂ Ω is the support of some permutation, we have Φh ≡ Φ for all Φ ⊆ Ω. But this is possible only if h has finite support, that is, h ≡ id.

Proposition 2.3 NS(Ω) induces an infinite cyclic group of outer automorphisms of G.

Proof. Since GNS(Ω), we see that NS(Ω) induces a group of automorphisms of G by conjugation. Since C_{NS(Ω)}(G) is

trivial, the outer automorphism group induced by NS(Ω) is NS(Ω)/G, which is isomorphic to Z.

Remark. Let Ω = {α_{i}: i < ω} be any enumeration of Ω. A typical generator of NS(Ω)/G is the equivalence class of the near-bijection σ: Ω → Ω defined by α_{i} ↦ α_{i+1} (for i < ω).

3 Structure of centralizers in G

In this section we give partial information on the structure of centralizers in G. From this, it will follow that Aut(G) induces an automorphism of B, and hence that Aut(G) fixes certain conjugacy classes of G.

For the first four lemmas we denote by ḡ a fixed element of G, and choose a fixed g ∈ S such that ḡ = gF S(Ω). We suppose that g has cycle type ∏(i^{n_{i}}: i ∈ T), where T ⊆ N ∪ {ω}. We define an equivalence relation F on Ω whose equivalence classes are listed below:

• The union of all the infinite cycles of g is an F-class F_{ω}.

• If 1 ≤ i < ω and g has infinitely many i-cycles, then the union of the i-cycles of g is an F-class F_{i}.

• The remaining points of Ω form a single F-class F_{0}, called the dregs.

We define the strong centralizer of ḡ in G, to be

SC_{G}(ḡ) := {h̄ ∈ C_{G}(ḡ): h fixes each F-class for some representative h of h̄}.

Note that, even though F depends on g, this definition does not depend on the choice of g. For, let g′ be another preimage of ḡ, and let F′ be the corresponding equivalence relation. Take h ∈ S such that h̄ ∈ SC_{G}(ḡ). Now, F′ and F differ only on a finite set Δ of points. There is a finitary permutation k fixing each F-class such that hk ∈ S_{(Δ)}. Then hk fixes each F′-class.

Lemma 3.1 Let g, ḡ be as above. Then the group C_{G}(ḡ)/SC_{G}(ḡ) is free abelian (where the trivial group is regarded as free abelian).

Proof. Let {F_{i}: i ∈ I} be the set of F-classes, and let K := ⨁(Zf_{i}: i ∈ I) be free abelian with {f_{i}: i ∈ I} as basis. We define a homomorphism χ: C_{G}(ḡ) → K as follows: if k̄ ∈ C_{G}(ḡ) then for each i ∈ I, we define

χ_{i}(k̄) = |F_{i}k \ F_{i}| − |F_{i} \ F_{i}k|.

Then χ_{i}(k̄) is well-defined, that is, it does not depend on the choice of the preimages g and k. Also put

χ(k̄) := ∑_{i∈I} χ_{i}(k̄)f_{i}.

Now g and k^{−1}gk agree on all but finitely many points, so k fixes all but finitely many F-classes and χ_{i}(k̄) = 0 for all but finitely many i ∈ I, so χ(k̄) does indeed lie in the direct sum K. It can be checked that χ is a homomorphism, its image is a subgroup of a free abelian group so is free abelian, and its kernel is SC_{G}(ḡ).

Problem 3.1 Find an expression for the rank of C_{G}(ḡ)/SC_{G}(ḡ), in terms of the cycle type of g.

We will need a considerable amount of information about all centralizers in G. To find C_{G}(ḡ), we must describe all elements h ∈ S such that [g, h] ∈ F S(Ω). We list five types of such elements, and then show that the images in G of elements of such types generate C_{G}(ḡ).

1. Obvious Centralizer Type These are elements of C_{S}(g). Conjugation by these elements fixes g. Note that C_{S}(g) ≅ ∏_{i∈T} (C_{i} Wr S_{n_{i}}), where C_{i} denotes the cyclic group of order i.

2. Finite/Finite Type These elements convert one finite set of finite cycles of g into another. Let A, B be the unions of supports of finitely many finite cycles of g, where

• |A| = |B| > 0;

• No cycle in B has the same length as one in A;

• A or B contains the support of a g-cycle of length i only if n_{i} = ω, that is, g has infinitely many cycles of length i.

Let χ: A → B be a bijection and define a permutation h as follows:

• h agrees with χ on A;

• If A contains an i-cycle, then h maps F_{i} \ A to F_{i}, respecting the cycle structure;

• If B contains an i-cycle, then h maps F_{i} to F_{i} \ B, respecting the cycle structure;

• h fixes all elements of all other F-classes.

Then g^{h} agrees with g except on the finite set B, so [g, h] ∈ F S(Ω).

3. Finite/Infinite Transfers Conjugation by these elements moves a finite cycle of g into an infinite cycle. Let (… a_{−2}a_{−1}a_{0}a_{1}a_{2} …) be an infinite cycle of g. Let (b_{1}b_{2} … b_{k}) be a finite cycle of g from the infinite F-class F_{k}. Define a permutation h as follows:

• h fixes a_{i} for i ≤ 0;

• a_{i}h = b_{i} for 1 ≤ i ≤ k; • a_{i}h = a_{i−k} for i > k;

• h maps F_{k} to F_{k} \ {b_{1}, b_{2}, …, b_{k}}, respecting the cycle structure;

• h fixes all elements of all other cycles.

Then g^{h} agrees with g except on {a_{0}, b_{k}}, and so [g, h] ∈ F S(Ω).

We refer to the inverses of such elements as infinite/finite transfers.

4. Infinite/Infinite Transfer These are elements which move a single point from one infinite cycle of g into another. Let

(… a_{−2}a_{−1}a_{0}a_{1}a_{2} …) and (… b_{−2}b_{−1}b_{0}b_{1}b_{2} …)

be two infinite cycles of g, and let h be the permutation of Ω whose only non-trivial cycle is

(… b_{−2}b_{−1}b_{0}a_{0}a_{1}a_{2} …).

Then the conjugate g^{h} differs from g only in that a_{0} lies in the b-cycle of g^{h}. Hence g^{h} agrees with g except on the set {a_{−1}, a_{0}, b_{0}}, so [g, h] ∈ F S(Ω).

5. Infinite/Infinite Recombination These are elements which cut two infinite cycles of g in half, then reattach the left side of one to the right side of the other. Let g have two infinite cycles as above, and let

h := (a_{1}b_{1})(a_{2}b_{2}) · · · .

Then g^{h} agrees with g except on {a_{0}, b_{0}}, so [g, h] ∈ F S(Ω).

Remark. Elements of types 1,4 and 5 lie in SC_{G}(ḡ), but elements of types 2 and 3 do not.

Proposition 3.2 (a) The group SC_{G}(ḡ) is generated by elements of G represented by elements of S of types 1,4 and 5.

(b) The group C_{G}(ḡ) is generated by elements of G represented by the above types.

Proof. (a) Let k ∈ S be such that [k, g] ∈ F S(Ω), and assume that k fixes setwise each F-class. We say that two infinite g-cycles

(… a_{−2}a_{−1}a_{0}a_{1}a_{2} …)

and

(… b_{−2}b_{−1}b_{0}b_{1}b_{2} …)

are initially equal if there are integers M and m such that a_{i} = b_{i−m} for all i < M. Finally equal is defined similarly.

For all but finitely many of the infinite g-cycles, its conjugate under k is also a g-cycle. Let (… a_{−2}a_{−1}a_{0}a_{1}a_{2} …) be any infinite g-cycle. Then

(… a_{−2}k, a_{−1}k, a_{0}k, a_{1}k, a_{2}k, …)

is initially equal to some g-cycle, and finally equal to some, perhaps different, gcycle. Then we can use infinite/infinite recombinations to reduce to the case where (a_{i}k) is initially and finally equal to a particular infinite g-cycle. We then use a series of infinite/infinite transfers, together with finitary permutations to reduce to the case where k preserves the structure of all the infinite cycles. Using further finitary permutations, we may suppose that k preserves all finite g-cycles, and it follows that k̄ ∈ C_{G}(ḡ).

(b) Let k ∈ S with [k, g] ∈ F S(Ω). Observe that F k^{△}F is finite for every Fclass F, and that all but finitely many of the F-classes are fixed setwise by k. By adjusting k by a finitary permutation, we can reduce to the case where, for each F-class F, either F ⊆ F k or F k ⊆ F. Note that since the dregs class F_{0} contains only finite cycles and has just finitely many cycles of any length, we must have F_{0}k = F_{0}.

Now consider the classes F_{i} where 1 ≤ i < ω. We wish to adjust k to ensure that F_{i}k = F_{i} for each such i, for then also F_{ω}k = F_{ω}. So fix i with 1 ≤ i < ω, and suppose that F_{i} ⊂ F_{i}k. (The case when F_{i} ⊃ F_{i}k is similar.) If g has an infinite cycle then we can use a series of infinite/finite transfers to move cycles of F_{i}k\F_{i} into F_{ω}, then a combination of finitary permutations on F_{ω} and finite/infinite transfers to move appropriate elements of F_{ω} back into F_{i}. If g has no infinite cycle, we multiply k by a single element of finite/finite type to reduce to the case where k fixes all F-classes setwise. We can then use a finitary permutation to ensure that k respects the cycle structure of finite cycles. After these reductions we have k ∈ SC_{G}(ḡ), so can apply part (a).

We now give a partial structure theorem for centralizers in G. We first define some normal subgroups of C_{G}(ḡ). Let g ∈ S be as above. Let

OC(ḡ) := C_{S}(g) · F S(Ω)/F S(Ω) = {h̄ : h ∈ S, [g, h] = 1}

the group of images of elements of obvious centralizer type. Also, define

N_{1}(ḡ) := {h̄ ∈ OC(ḡ) : h fixes setwise the support of each infinite g-cycle},

N_{2}(ḡ) := {h̄ ∈ SC_{G}(ḡ) : |Dh^{△}D| < ℵ_{0}, if D is the support of an infinite g-cycle},

and

N_{3}(ḡ) := ⟨N_{2}(ḡ), OC(ḡ)⟩.

These definitions are independent of the choice of the representative g.

Theorem 3.3 (a) The groups

N_{1}(ḡ), OC(ḡ), N_{2}(ḡ), N_{3}(ḡ) and SC_{G}(ḡ)

are each normal subgroups of C_{G}(ḡ), with

N_{3}(ḡ) ≤ SC_{G}(ḡ) ≤ C_{G}(ḡ).

(b) N_{2}(ḡ) ∩ OC(ḡ) = N_{1}(ḡ). Hence we have the isomorphisms N_{3}(ḡ)/ OC(ḡ) ≅ N_{2}(ḡ)/N_{1}(ḡ) and N_{3}(ḡ)/N_{2}(ḡ) ≅ OC(ḡ)/N_{1}(ḡ).

(c) (i) The groups C_{G}(ḡ)/SC_{G}(ḡ) and N_{2}(ḡ)/N_{1}(ḡ) are free abelian, and the latter has rank n if g has n + 1 infinite cycles (n finite) and ℵ_{0} if g has infinitely many infinite cycles.

(ii) The quotient OC(ḡ)/N_{1}(ḡ) is isomorphic to the symmetric group on the set of infinite cycles of ḡ.

(iii) The quotient SC_{G}(ḡ)/N_{3}(ḡ) is isomorphic to the finitary symmetric group on the set of infinite g-cycles.

(d) If g has more than one infinite cycle then the five normal subgroups listed in (a) are distinct, and if g has at most one infinite cycle then they are all equal.

Proof. All parts of this are easy, so we omit details. To show that N_{2}(ḡ)/N_{1}(ḡ) is free abelian, we argue much as in the proof of Lemma 3.1. This quotient is generated by infinite/infinite transfers, modulo N_{1}(ḡ). By Proposition 3.2(a), the finitary symmetric group SC_{G}(ḡ)/N_{3}(ḡ) is generated by infinite/infinite recombinations (modulo N_{3}(ḡ)).

We now define, for any h ∈ S, the group

G(h̄) := {k̄ : supp(k) ⊆ fix(h)}.

Observe that this definition is independent of the choice of the element h ∈ S such that h̄ = h · F S(Ω). Also, clearly G(ḡ)C_{G}(ḡ).

Lemma 3.4 (a) Suppose that g ∈ S has infinitely many fixed points. Then G(ḡ) is isomorphic to G, and is the only normal subgroup of C_{G}(ḡ) which is isomorphic to G.

(b) Suppose that g ∈ S has just finitely many fixed points. Then there is no normal subgroup of C_{G}(ḡ) isomorphic to G.

(c) Let g, h ∈ S and φ ∈ Aut(G), and suppose ḡφ = h̄. Then g has infinitely fixed points if and only if h has infinitely many fixed points, and if they do, then G(ḡ)φ = G(h̄).

Proof. (a) Let Γ := fix(g). Clearly

G(ḡ) = Sym(Γ) · F S(Ω)/F S(Ω) ≅ Sym(Γ)/F S(Γ) ≅ G.

Now let N̄ be any normal subgroup of C_{G}(ḡ) isomorphic to G. We shall show that N̄ = G(ḡ). First observe that G(ḡ) is simple, so for any normal subgroup M of C_{G}(ḡ), either G(ḡ) ≤ M or G(ḡ) ∩ M = 1. In particular, by Lemma 3.1, as N̄ is non-abelian it must lie in SC_{G}(ḡ). Also, by Theorem 3.3, as G is not embeddable in a free abelian group or a finite or finitary symmetric group, we must have N̄ ≤ OC(ḡ).

Define N := {h ∈ S : h̄ ∈ N̄}. Observe that

C_{S}(g) = ∏(C_{i} Wr S_{n_{i}} : i ∈ T),

and that 1 ∈ T and n_{1} is infinite, and that OC(ḡ) = C_{S}(g) · F S(Ω)/F S(Ω). Put

K := C_{1} Wr S_{n_{1}} × ∏(C_{i}^{n_{i}} : i ∈ T, i > 1),

and

L := {k ∈ S : supp(k) ⊆ fix(g)} ≅ S_{n_{1}} = S.

Then L ≤ K. Put K̄ := K · F S(Ω)/F S(Ω) and L̄ := L · F S(Ω)/F S(Ω). Then L̄ and K̄ are normal subgroups of OC(ḡ) and K̄/L̄ is abelian. The proof will be complete if we can show that N̄ ≤ L̄ (since L̄ is simple), so we shall suppose that this is false. It follows that N̄ ∩ K̄ = 1; for otherwise N̄ ≤ K̄ which is impossible as K̄/L̄ is abelian and N̄ is non-abelian and simple.

Now pick h ∈ N \ L · F S(Ω). Then h ∉ K · F S(Ω), so, up to a finitary permutation, h induces a permutation of infinite support on the set of non-trivial cycles of g. It is now easy to choose k ∈ K so that [h, k] ∉ L · F S(Ω). Now [h, k] ∈ N ∩ (K · F S(Ω) \ L · F S(Ω)), which contradicts the last paragraph.

Part (b) follows from the proof of uniqueness in (a), and (c) follows from (a) and (b).

We will adopt the notation G(ḡ), G(h̄) later in the paper. Also, from now on φ will denote an element of Aut(G).

Lemma 3.5 Let φ ∈ Aut(G), let ḡ, h̄, k̄ ∈ G, and assume that g, h, k each have infinitely many fixed points. Then

(a) supp(ḡ) = supp(h̄) if and only if supp(ḡφ) = supp(h̄φ),

(b) supp(ḡ) < supp(h̄) if and only if supp(ḡφ) < supp(h̄φ),

(c) supp(k̄) = supp(ḡ) ∧ supp(h̄) if and only if supp(k̄φ) = supp(ḡφ) ∧ supp(h̄φ).

Proof. To see (a), observe that supp(ḡ) ≤ supp(h̄) if and only if G(h̄) ≤ G(ḡ), and by Lemma 3.4(c), this holds if and only if G(h̄φ) ≤ G(ḡφ). But, again by Lemma 3.4(c), we have G(ḡφ) = G(ḡ)φ and G(h̄φ) = G(h̄)φ.

Part (b) follows directly from (a). Part (c) then follows since

supp(k̄) = supp(ḡ) ∧ supp(h̄)

if and only if supp(k̄) ≤ supp(ḡ) ∧ supp(h̄) and supp(ℓ̄) ≤ supp(k̄) for all ℓ̄∈ G such that supp(ℓ̄) ≤ supp(ḡ) ∧ supp(h̄).

It follows from this lemma that Aut(G) embeds in Aut(B) (via the action of each automorphism on the set of supports of elements with infinitely many fixed points). Observe that if supp(ḡ) ∧ supp(h̄) = 0 then there are representatives g of ḡ and h of h̄ with disjoint supports.

Next, we define the conjugacy classes in which we will be especially interested. For any subset T ⊆ N \ {0}, let

C(T)_{inf} := {ḡ : g ∈ S, g has cycle type 1^{ω} · ∏(n^{ω} : n ∈ T)}.

Also, define

C(T) := {ḡ : g ∈ S, g has cycle type 1^{ω} · ∏(n^{1}: n ∈ T)}.

Both C(T)_{inf} and C(T) are conjugacy classes of G. Our goal is to show that they are preserved by any automorphism of G.

For T = {2} we shall write C(2)_{inf} for C({2})_{inf}.

Lemma 3.6 Let φ ∈ Aut(G), and let T ⊆ N \ {0}, and ḡ ∈ C(T)_{inf}. Then ḡφ ∈ C(T)_{inf}.

Proof. First, observe that for any i with 1 < i < ω and h ∈ S with infinitely many fixed points, h has infinitely many i-cycles if and only if

supp(h̄^{i}) < ⋀(supp(h̄^{j}) : 1 ≤ j < i, j|i).

By Lemma 3.5, it follows that if k ∈ S with k̄ = ḡφ, then for each i with 1 < i < ω, the element k has infinitely many i-cycles if and only if i ∈ T.

Next, again assuming h ∈ S has infinitely many fixed points, note that h has a single infinite cycle and just finitely many non-trivial finite cycles if and only if supp(h̄) ≠ 0 and we cannot write h̄ = h̄_{1}h̄_{2} with the supports of h̄_{1} and h̄_{2} disjoint and not equal to 0 or 1 ∈ B. Hence, given such an element h, there is k ∈ S with a single infinite cycle and no other non-trivial cycles such that k̄ = h̄φ. Also, if h ∈ S is not of this form, then h has at least one infinite cycle if and only if there are h_{1}, h_{2} ∈ S such that h̄ = h̄_{1}h̄_{2}, the h̄_{i} have disjoint supports not equal to 0 or 1 in B, and h_{1} has a single infinite cycle and no other non-trivial cycles. It follows from Lemma 3.5 that if k ∈ S with k̄ = ḡφ, then k has no infinite cycles. In fact, this argument shows that if k_{1}, k_{2} ∈ S and k̄_{1}φ = k̄_{2} and k_{1} has infinitely many fixed points, then k_{1} and k_{2} have the same number of infinite cycles.

Finally, suppose that h ∈ S has infinitely many fixed points and no infinite cycles. Then the dregs class F_{0} of h is finite if and only if, whenever there are h_{1}, h_{2} ∈ S such that h̄ = h̄_{1}h̄_{2} with h̄_{1}, h̄_{2} having disjoint non-trivial supports and h_{1} having finitely many i-cycles for all i (1 < i < ω), the elements h̄_{2} and h̄ are conjugate in G. Again, this condition is φ-invariant, so the lemma follows from this and the previous two paragraphs.

Lemma 3.7 Let φ ∈ Aut(G), and let T be an infinite set of nontrivial finite cycle lengths. Then C(T) is φ-invariant.

Proof. We will show that C(T) is the only conjugacy class of G \ {1} whose elements ḡ satisfy all the following properties.

(a) there is k̄ ∈ C(T)_{inf} with supp k̄ ∧ supp ḡ = 0 such that k̄ḡ ∈ C(T)_{inf};

(b) for any moiety R of T, there is no k̄ ∈ C(R)_{inf} with supp k̄ ∧ supp ḡ = 0 and k̄ḡ ∈ C(R)_{inf};

(c) if ḡ = h̄k̄ where h̄, k̄ ∈ G\{1}, supp h̄∧supp k̄ = 0 and supp h̄∨supp k̄ = supp ḡ, then one of (a) or (b) fails when ḡ is replaced by h̄.

By Lemma 3.5, the conditions on supports are preserved by φ, and by Lemma 3.6, C(T)_{inf} is φ-invariant, so the conditions (a)-(c) are preserved by φ. Hence, given the above, C(T) is φ-invariant, as required.

Clearly if ḡ ∈ C(T) then it satisfies (a)–(c). Conversely, assume that ḡ ∈ G \ {1} satisfies (a)–(c), and choose a representative g. Property (a) says that g has infinitely many fixed points, that all but finitely many of the cycles of g have lengths lying in T, and that g has no infinite cycles. Property (b) says that all but finitely many elements of T arise as lengths of cycles of g. Thus (a) and (b) imply that the set of cycle lengths of g is almost exactly T. If the cycle type of g differs more than finitely from 1^{ω} · ∏(n^{1}: n ∈ T), then (c) fails when we let h be the product of exactly one n-cycle of g for each n ∈ T, and let k be the product of the remaining g-cycles.

4 Proof of the Theorem

In this section, we prove Theorem 1.1, that is, that every outer automorphism of G is one of the outer automorphisms induced by a near symmetry as described in Section 2.

Let φ be any automorphism of G. We must find f_{φ} ∈ NB(Ω) such that the action of φ on G is that of conjugation by f_{φ}. In the introduction, we defined the support of an element of G, and in

Lemma 3.5 we showed that φ respects supports of elements of G and the ‘containment’ relation induced on supports by the subset relation on Ω. We also showed that certain conjugacy classes of G in which we are interested are respected by φ, that is, are not fused by φ. We now pick a particular conjugacy class C(T) of G, a subset D of unordered pairs from C(T), and an equivalence relation E on D, and show that C(T), D and E are invariant under φ. Elements of D/E will be naturally identified with ordered moieties of Ω, and this enables us to define f_{φ} ∈ NB(Ω) corresponding to φ. We show that f_{φ} and φ agree in their action on one non-trivial conjugacy class of G. Then since G is a simple group, this conjugacy class generates G and so the actions of f_{φ} and φ agree on the whole of G, as required.

Fix a set T = {n_{i}: i < ω} of finite nontrivial cycle lengths such that for all i < ω we have n_{i} > 2 ∑_{j<i} n_{j}. We note without proof the following easy lemma.

Lemma 4.1 Let T be as above. Then

(a) if a, b, c ∈ T then c ≠ 2a + b − 2;

(b) if a, b, c ∈ T then 2a ≠ 2b + c − 1;

(c) if a, b ∈ T then a ≠ 2b − 1;

(d) if a, b, c, d ∈ T with a + b = c + d then {a, b} = {c, d};

(e) if a, b, c, d ∈ T with 2a + b = 2c + d then a = c and b = d.

We say that {g, h} is a matched C(T)-pair if:

(a) g, h have cycle types 1^{ω} · ∏(i^{1}: i ∈ T);

(b) supp(g) ∪ supp(h) is a moiety of Ω;

(c) for each n ∈ T, the n-cycle of g meets supp(h) in a singleton lying in the n-cycle of h.

Next, let D be the set of all unordered pairs {ḡ, h̄} for which there are corresponding elements g, h ∈ S which form a matched C(T)-pair. We say that {g, h} is a matched cover of {ḡ, h̄}.

If {ḡ, h̄} ∈ D, we will write supp(g, h) for supp(g) ∪ supp(h). If {g, h} is a matched cover of {ḡ, h̄}, let the spine σ(g, h) of {g, h} be the ω-sequence from Ω whose i^{th} entry is the point in which the n_{i}-cycles of g and h intersect. Then the ribcage of {g, h} is the set supp(g, h) \ (supp(g) ∩ supp(h)), that is the elements of the supports of g and h which are not in the spine.

Lemma 4.2 The set D defined above is φ-invariant for any φ ∈ Aut(G).

Proof. We show that if ḡ, h̄ ∈ G then {ḡ, h̄} ∈ D if and only if the following conditions all hold:

(a) ḡ, h̄ ∈ C(T);

(b) supp(ḡ) ∨ supp(h̄) ≠ 1;

(c) for all k̄ ∈ C(2)_{inf} such that supp(k̄) = supp(ḡ) ∧ supp(h̄) we have ḡk̄, h̄k̄ ∈ C(Q) for some subset Q of {n_{i} + n_{j}: n_{i}, n_{j} ∈ T };

(d) ḡh̄ ∈ C(R), where R = {2n − 1: n ∈ T }.

Here, part (b) says that supp(g) ∪ supp(h) is a moiety of Ω. Part (c) ensures that for some g, h ∈ S representing ḡ, h̄, every non-trivial cycle of g intersects supp(h) in at most a singleton, and similarly with g, h interchanged. For, if the n_{i}-cycle of g intersects supp(k) in a singleton, which k pairs with a singleton from the n_{j}-cycle of g, then gk contains an (n_{i} +n_{j})-cycle. On the other hand, if, say, there are infinitely many cycles of g which each intersect supp(h) in more than one point, then k can be chosen so that gk contains an infinite cycle. We leave the details to the reader.

If the m-cycle of g meets the k-cycle of h in a singleton, then they form an (m+k−1)-cycle in the product gh. By Lemma 4.1(d), if m, k, n ∈ T then m+k−1 = 2n−1 if and only if m = k = n. Thus part (d) ensures that these cycles are matched up correctly, that is, the n-cycle of g meets the n-cycle of h.

By Lemmas 3.5, 3.6 and 3.7, the conditions in (a) to (d) are preserved by any automorphism. So D is φ-invariant.

Next, we define an equivalence relation E on D. If {ḡ_{1}, h̄_{1}}, {ḡ_{2}, h̄_{2}} ∈ D, then we say that they are equivalent under E if they have matched covers {g_{1}, h_{1}} and {g_{2}, h_{2}} respectively, such that σ(g_{1}, h_{1}) = σ(g_{2}, h_{2}).

Lemma 4.3 Let φ ∈ Aut(G). Then the equivalence relation E is φ-invariant.

Proof. We say that g_{1}, g_{2}, g_{3}, g_{4} ∈ S form a matched quadruple if

(a) {ḡ_{i}, ḡ_{j}} ∈ D and has {g_{i}, g_{j}} as a matched cover for all 1 ≤ i < j ≤ 4;

(b) all sequences σ(g_{i}, g_{j}), (1 ≤ i < j ≤ 4) are equal;

(c) ⋃{supp(g_{i}): 1 ≤ i ≤ 4} is a moiety of Ω;

Claim: The set of matched quadruples in G is φ-invariant.

Proof of Claim: Note that {ḡ_{1}, ḡ_{2}, ḡ_{3}, ḡ_{4}} is a matched quadruple if and only if {ḡ_{i}, ḡ_{j}} ∈ D for all 1 ≤ i < j ≤ 4 and supp(ḡ_{i})∧supp(ḡ_{j}) = ⋀{supp(ḡ_{k}): 1 ≤ k ≤ 4} and ⋁{supp(ḡ_{k}): 1 ≤ k ≤ 4} ≠ 1. These conditions are clearly preserved by φ, which yields the claim.

Now take {ḡ_{1}, h̄_{1}}, {ḡ_{2}, h̄_{2}} ∈ D such that {ḡ_{1}, h̄_{1}}E{ḡ_{2}, h̄_{2}}. Let {g_{1}, h_{1}} and {g_{2}, h_{2}} be matched covers. We can find {ḡ_{3}, h̄_{3}} ∈ D and {ḡ_{4}, h̄_{4}} ∈ D with matched covers {g_{3}, h_{3}} and {g_{4}, h_{4}} such that

supp(g_{1}, h_{1})∪supp(g_{3}, h_{3}), supp(g_{3}, h_{3})∪supp(g_{4}, h_{4}) and supp(g_{2}, h_{2})∪supp(g_{4}, h_{4})

are moieties, the spines σ(g_{i}, h_{i}) all agree and the ribcages of supp(g_{1}, h_{1}) and supp(g_{3}, h_{3}) are disjoint, as are the ribcages of supp(g_{3}, h_{3}) and supp(g_{4}, h_{4}), and the ribcages of supp(g_{4}, h_{4}) and supp(g_{2}, h_{2}). Then each of {ḡ_{1}, h̄_{1}, ḡ_{3}, h̄_{3}}, {ḡ_{3}, h̄_{3}, ḡ_{4}, h̄_{4}} and {ḡ_{4}, h̄_{4}, ḡ_{2}, h̄_{2}} is a matched quadruple of G. Hence, by the claim, their images under φ are all matched quadruples of G. Thus

{ḡ_{1}φ, h̄_{1}φ}E{ḡ_{3}φ, h̄_{3}φ}E{ḡ_{4}φ, h̄_{4}φ}E{ḡ_{2}φ, h̄_{2}φ},

as required.

It follows from the last lemma that φ has a natural induced action on the set of almost-equality classes of ω-sequences which enumerate moieties of Ω. We are now in a position to construct a near-bijection f_{φ} corresponding to φ. Let Ω = Ω_{0} ∪ Ω_{1} be a partition of Ω into moieties, and let {ḡ, h̄}, {k̄, l̄} ∈ D have matched covers {g, h}, {k, l} respectively such that σ(g, h) is an enumeration of Ω_{0} and σ(k, l) is an enumeration of Ω_{1}. Since φ preserves the Boolean algebra B, we see that if {g′, h′} and {k′, l′} are matched covers of {ḡφ, h̄φ} and {k̄φ, l̄φ} respectively, then σ(g′, h′) and σ(k′, l′), regarded as sets, have finite intersection and their union is cofinite in Ω. There is a near-bijection f_{φ} of Ω which induces a map taking σ(g, h) to σ(g′, h′) and taking σ(k, l) to σ(k′, l′) (except on a finite set).

The next lemma shows essentially that this action preserves subsequences. (Note that these subsequences are not necessarily increasing subsequences; that is, they may involve permutations of the elements of the sequence, so may not induce the same ordering.)

Lemma 4.4 Let φ ∈ Aut(G). Suppose that {ḡ, h̄}, {k̄, l̄} ∈ D have matched covers {g, h}, {k, l} respectively, and that σ(g, h) meets supp(k) ∩ supp(l) in a subsequence of σ(k, l), that is, there is A ⊆ ω and an injection χ: A → ω such that if i ∈ A then the i^{th} element of σ(g, h) is the χ(i)^{th} element of σ(k, l), and if i ∉ A then the i^{th} element of σ(g, h) does not lie in σ(k, l). Then there are matched covers {g′, h′} and {k′, l′} of {ḡφ, h̄φ} and {k̄φ, l̄φ} respectively such that if i ∈ A then the i^{th} element of σ(g′, h′) is the χ(i)^{th} element of σ(k′, l′), and if i ∉ A then the i^{th} element of σ(g′, h′) does not occur in σ(k′, l′).

Proof. By replacing (k̄, l̄) by a pair E-equivalent to it, and using Lemma 4.3, we may suppose that

supp(k) ∩ supp(l) ∩ supp(g) ∩ supp(h) = supp(g, h) ∩ supp(k, l).

It follows that we may choose matched covers {g′, h′} of {ḡφ, h̄φ} and {k′, l′} of {k̄φ, l̄φ} so that

supp(k′) ∩ supp(l′) ∩ supp(g′) ∩ supp(h′) = supp(g′, h′) ∩ supp(k′, l′).

Hence there is a subset B of ω and an injection χ′ : B ⟶ ω such that if i ∈ B then the i^{th} element of σ(g′, h′) is the χ′(i)^{th} element of σ(k′, l′), and if i ∉ B then the i^{th} element of σ(g′, h′) is not in σ(k′, l′), for almost all i ∈ ω. We must show that A = B and χ = χ′ almost everywhere (for then, by finite adjustments of g′, h′, k′, l′, we may arrange that equality holds everywhere).

It is clear that ghk has cycle type C(S) where

S = {2n_{i} − 1 : i ∉ A} ∪ {2n_{i} + n_{χ(i)} − 2 : i ∈ A} ∪ {n_{i}: i ∉ χ(A)}.

Similarly, g′h′k′ almost has cycle type

S = {2n_{i} − 1 : i ∉ B} ∪ {2n_{i} + n_{χ′(i)} − 2: i ∈ B} ∪ {n_{i}: i ∉ χ′(B)}.

It follows by Lemma 4.1(a),(b),(c) and (e) that A^{△}B is finite and χ(i) = χ′(i) for almost all i ∈ ω, as required.

Lemma 4.5 Let f_{φ} be as defined above. Then the ≡-class of f_{φ} is independent of the choice of Ω_{0}, Ω_{1} and of ḡ, h̄, k̄, l̄.

Proof. Pick another partition Ω = Ω_{0}^{∗}∪Ω_{1}^{∗} into moieties, and let {ḡ^{∗}, h̄^{∗}}, {k̄^{∗}, l̄^{∗}} ∈ D have matched covers {g^{∗}, h^{∗}}, {k^{∗}, l^{∗}} so that σ(g^{∗}, h^{∗}) is an enumeration of Ω_{0}^{∗} and σ(k^{∗}, l^{∗}) is an enumeration of Ω_{1}^{∗}. Define a corresponding near-bijection f_{φ}^{∗} of Ω. We must show f_{φ} ≡ f_{φ}^{∗}. This, however, follows almost immediately from Lemma 4.4.

To complete the proof of the theorem, we must prove the following lemma.

Lemma 4.6 Let f_{φ} be as above, and let f̄_{φ} be the element of NS(Ω) corresponding to f_{φ}. Then for all ḡ ∈ C(2)_{inf} we have f̄_{φ}^{−1}ḡf̄_{φ} = ḡφ.

Proof. Let g := ∏_{i∈ω}(a_{2i}a_{2i+1}). Choose permutations x, y ∈ S such that {x̄, ȳ} ∈ D, the set {x, y} is a matched C(T)-pair, and such that σ(x, y) is the sequence (a_{i} : i ∈ ω). Then supp(ḡ) = supp(x̄)∧supp(ȳ), so supp(ḡφ) = supp(x̄φ)∧supp(ȳφ). Also, {x̄φ, ȳφ} ∈ D, so we may choose a corresponding matched C(T)-pair {x′, y′}. Let σ(x′, y′) = (b_{i} : i < ω). Then by Lemma 4.5, a_{i}f_{φ} = b_{i} for all but finitely many i ∈ ω. In particular, f_{φ}^{−1}gf_{φ} and ∏_{i<ω}(b_{2i}b_{2i+1}) are ≡-equivalent. Also, by Lemma 3.6 (or a direct argument), ḡφ ∈ C(2)_{inf}, and if g′ ∈ S with ḡ′ = ḡφ, then the symmetric difference supp(g′)^{△}{b_{i} : i ∈ ω} is finite. It remains to show that g′ and ∏_{i∈ω}(b_{2i}b_{2i+1}) differ by a finitary permutation.

Let u := gx. Then u has cycle type 1^{ω} ∏((n_{2i} + n_{2i+1})^{1} : i < ω). This says that, for each i < ω, g pairs up the n_{2i}-cycle of x with the n_{2i+1}-cycle of x. By Lemma 3.7, ūφ and ū are conjugate in G, so have preimages in S with cycle types differing finitely. If g′ induces the transposition (b_{i}b_{j}) then x′g′ has an (n_{2i} +n_{2i+1})- cycle. It now follows from Lemma 4.1(d) that g′ differs finitely from ∏_{i∈ω}(b_{2i}b_{2i+1}), as required.

Proof of Theorem 1.1. The main theorem follows immediately from Lemma 4.6. For, since the conjugacy class C(2)_{inf} generates G, conjugation by f̄_{φ} induces the action of φ on all elements of G.

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